Collateral Risk, Repo Rollover and Shadow Banking

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Collateral Risk, Repo Rollover and Shadow Banking
Shengxing Zhangú†
Department of Economics, New York University
[download the latest version]
January 26, 2014
Abstract
I build a dynamic model of the shadow banking system and the interbank repo market to
understand their efficiency and stability. The model emphasizes a key friction: the maturity
mismatch between short-term repo and long-term investments that banks need to finance. The
haircut, interest rate, and default rate of the repo contract are endogenously determined in the
model, as are the volume of lending and liquidity hoarding. Default is shown to be contagious.
Finally, when collateral risk increases unexpectedly, the haircut and interest rate overshoot,
triggering massive defaults and persistently hiking the default rate and depressing investment.
ú
†
email: [email protected]
I am indebted to Ricardo Lagos, Douglas Gale, Boyan Jovanovic and Thomas Sargent for their support and
discussions. I am especially grateful to Ricardo Lagos and Douglas Gale for their guidance on the project. I also
thank David Andolfatto, Gara Afonso, Saki Bigio, Dan Cao, Katarína Borovička, Jaroslav Borovička, Emmanuel
Farhi, Klaus-Peter Hellwig, Ben Lester, Yaron Leitner, Jesse Perla, Edouard Schaal, Cecilia Parlatore Siritto, Chris
Tonetti, Laura Veldkamp, Venky Venkateswaran, Gianluca Violante and all other participants of the Sargent Research
Group, the Student Macro Lunch seminar of the Department of Economics at New York University, the Macro Lunch
seminar at NYU Stern, the student workshop at Wharton and the brown bag seminar at the Federal Reserve Bank
of St. Louis.
1
1
Introduction
The shadow banking system is an essential part of the process of credit creation in modern banking.1
The process relies heavily on short-term debt instruments such as repurchase contracts (repo), a
short-term collateralized debt contract with safe harbor provisions.2 Just before the great recession,
the gross volume of outstanding repo contracts reached $10 trillion in both the US and Euro-zone
repo markets. To put this amount into perspective, it amounted about 70% of GDP in the respective
areas in 2007.3
In 2007, the risk on collateral assets increased unexpectedly due to the sharp decline in housing
prices.4 Concerned about the quality of collateral assets, financial intermediaries reduced their
repo exposure to each other and began hoarding liquidity. The haircut and interest rates shot up.
This eventually led to the downfall of Lehman Brothers, which ran out of resources to finance its
long-term investments.5
The crisis in the repo market and the shadow banking system exposed the instability of the
system and left us with the following questions. What is the source of systemic risk in the shadow
banking system? Are the efficiency and stability of the system affected by frictions in the repo
market? What triggered the crisis? Why has the system not fully recovered even five years after
the outbreak of the crisis?
To answer these questions, I build a dynamic model of the shadow banking system and the
interbank repo market. Banks have an initial endowment of cash and collateral. Each bank
anticipates an investment option that arrives according to a Poisson process. While a bank waits
for its investment option to arrive, it lends through the repo market to banks that already have an
1
See Pozsar et al. [2013] for the credit intermediation process in the shadow banking system. And I will explain
its institutional features in more detail in the next section.
2
In legal terms, a repo contract is a combination of two outright transactions, sales at the moment the contract
is signed and purchase at a future date at a price according to the contract. Since it can be interpreted either as a
combination of two spot trades or as a secured loan, it helps some financial institutions circumvent legal restrictions
to lending to other institutions or to carrying out spot trade. Another difference between the repo contract and a
secured loan is that when a borrower defaults, the collateral asset is not subject to automatic stay. The safe harbor
provision makes financing through a repo contract popular. See Garbade [2006] for more details.
3
See Hördahl and King [2008].
4
ABX indices, price index for CDS over a collection of mortgage backed securities, dropped.
5
Gorton [2009] gives a detailed description of the unfolding of the events in the crisis.
2
investment opportunity. The model emphasizes a key friction: the maturity mismatch between the
short-term repo and the long-term investment that banks need to finance. The maturity mismatch
results in borrowers having to roll over their debt until the investment matures and they are able to
repay their loan. If the investment does not mature before the borrower reaches his (endogenous)
debt limit, the borrower will be forced to default.
The haircut, interest rate and default rate of the repo contract are all endogenously determined
in the model. As we will see, the endogenous haircut and default rate allow me to study how
changes in the primitives of the model affect the borrowing constraint and the externalities caused
by default.
I show that systemic risk arises because default triggers more default. When a borrower defaults,
the lender’s portfolio becomes less liquid: she gains a collateral asset but loses her claim to future
cash from her counterparty. Then, when the lender’s investment opportunity arrives, she relies
more on secured borrowing (backed by the collateral assets) and less on her own cash. This makes
her more likely to default because, other things being equal, she reaches her debt limit faster. Thus,
counterparty default is contagious.
To understand the failure of the shadow banking system and to check the robustness of the
theory, I extend the model to allow for collateral risk6 and find that an increase in collateral
risk increases counterparty default risk. When collateral risk increases, the repo market dries up
through two channels, liquidity hoarding and counterparty default. Banks reduce funding to the
repo market and hoard more liquidity to secure funding for their own investment in the future.
With less funding in the repo market, the equilibrium haircut and interest rate increase, the debt
limit is reached faster, and default is more likely. Counterparty default transforms the lenders’
portfolios and further decreases the supply of funding to the repo market. So, as collateral risk
increases, the efficiency of the financial system declines.
I use the dynamic model to study both the efficiency of the shadow banking system in the steady
state and the stability of the system in response to an unanticipated shock. The stability of the
system can be measured by two metrics: the magnitude of the initial response and the persistence
6
At the time, 50% of primary broker-dealers’ repo contracts are backed by such less-liquid securities as corporate
securities, mortgage-backed securities and other asset-backed securities, and 65% of the contracts are overnight. See
Adrian et al. [2009].
3
of the efficiency loss on the transition path to the new steady state. My objective is to understand
the stability of the financial system at the onset of the great recession. To do this, I characterize the
transition dynamics triggered by an unexpected increase in collateral risk. On impact, the hoarding
motive of lenders imposes a downward pressure on the supply of funds to the repo market. As a
result, both the haircut and interest rate increase sharply to clear the market. This leads to a
tightening of the borrowers’ debt limit and a massive default by those borrowers who suddenly
find themselves over their debt limit. In addition, the rest of the borrowers who started borrowing
before the crisis now face a debt overhang problem, which increases the default rate from that point
onward. Since counterparty default is contagious, the massive initial default and the increase in the
default rate have a persistent effect on the equilibrium path: Repo lending and investment remain
low, and the default rate remains high for an extended period of time. The systemic risk from
contagious counterparty default increases both the magnitude and the persistence of the efficiency
loss on the dynamic transition path to a new steady state.
Literature review
Previous research has emphasized asymmetric information and market failure as causes of the
financial crisis (Chiu and Koeppl [2011], Camargo and Lester [2011]). Dang et al. [2009, 2012],
Gorton and Ordonez [2012], Farhi and Tirole [2012] and Hellwig and Zhang [2012] study the effect
of endogenous information structure and market liquidity. This paper takes a different approach,
exploring the possibility that there is a simpler explanation for the crisis – namely, an increase in
collateral risk amplified through the mechanism of contagious counterparty default.
This approach is closely related to that of Kocherlakota [2001], in which collateral risk makes
it harder for a lender to enforce payment of the promised share of a project. As in Kocherlakota
[2001], it is too costly to collect the promised payment from borrowers, other than to seize the
risky collateral. The difference is that, when the value of collateral drops, lenders are not able to
withdraw funding from borrowers, even though only a tiny fraction of lenders observe the shock.
The repayment of repo contracts is supported mostly by debt rollover. But equilibrium rollover
will collapse when a small fraction of lenders want to withdraw, after observing the shock.
The term structure of repo borrowing in my model is exogenous. Brunnermeier and Oehmke
[2013] shows that the exemption from automatic stay of short-term repo contracts triggers a matu4
rity rat race, so, in equilibrium, banks borrow inefficiently short-term. But it would be interesting
to endogenize the term structure in the future and study its effect, as in Williamson [2013].
The repo market in my model allows banks and investors to share profitable opportunities, as
in Kiyotaki and Moore [2002] and Berentsen et al. [2007]. Many authors have used the workhorse
model of Diamond and Dybvig [1983] to study traditional banking crises, but few have provided
models of the repo market. Martin et al. [2011] focuses on the repo market between cash providers
and financial intermediaries. I focus on repo lending between financial intermediaries to provide a
complementary approach to understanding systemic risk. Gennaioli et al. [2013] presents another
model of the shadow banking system, in the spirit of Diamond and Dybvig [1983]. In empirical work
by Gorton and Metrick [2012a], the authors take the view that the recent crisis was a system-wide
self-fulfilling bank run. While the Diamond-Dybvig model is important for understanding panics,
it is unclear how it explains the ‘runs’ that occurred in the bilateral repo market. Collateral plays
the same role in repo markets that deposit insurance plays in traditional banking and should have
prevented a self-fulfilling bank run in a Diamond-Dybvig-style model. I focus, instead, on studying
systemic risk and equilibrium dynamics in the repo market without runs.
This paper subscribes to the view, expressed in Sargent [2013], that liquidity problems are the
result of market incompleteness, and, hence, the solution to a liquidity problem is model-dependent.
I model the shadow banking sector with two questions from Moore [2011] in mind: Why do financial
intermediaries hold mutual gross positions, and do these gross positions create systemic risk?
The paper is related to the study of banks’ risk-taking behavior, as in Allen and Gale [2001]. In
my paper, investment in a profitable long-term project is risky, as it is uncertain when the project
will mature.
In my paper, a lender whose counterparty defaults is more likely to default when she invests in
her long-term project. The financial contagion effect in the paper is in the spirit of Allen and Gale
[2000]. Here, the effect takes place on the dynamic equilibrium path and has explicit implications
for variables such as the haircut.
Liquidity hoarding that results from collateral risk is reminiscent of the precautionary demand
for funding, as in Frenkel and Jovanovic [1980]. There is also a speculative motive for liquidity
hoarding – to buy assets at fire sale prices in the future – as in Gale and Yorulmazer [2013]. Gale
and Yorulmazer [2013] list two possible explanations for the phenomenon of liquidity hoarding:
5
counterparty default risk and the fear that participation in lending may compromise a lender’s
future access to liquidity. Both ingredients contribute to the repo market freeze and to liquidity
hoarding in my model.
The (endogenous) characterization of haircuts is related to the study of endogenous leverage initiated by Geanakoplos and Zame [1997] and developed by Fostel and Geanakoplos [2012]. Haircuts
in these papers are pinned down by the price of Arrow securities, subject to additional constraints.
In my model, the repo contract is the only contract traded in the market, and the haircut is
determined by a necessary condition for equilibrium rollover. This complements the extensive literature on credit cycles, starting with Kiyotaki and Moore [1997a] and Bernanke and Gertler [1989],
with recent development including Adrian and Shin [2010], Brunnermeier and Sannikov [2012] and
Gertler and Kiyotaki [2013]. In all of these papers on credit cycles, the borrowing constraint is
exogenous and always binding.
Another feature of the repo market is that investors can build leverage through rehypothecation
(see Singh and Aitken [2010] for empirical evidence; see Bottazzi et al. [2012] for a theory of rehypothecation). Additional issues introduced to the repo market by rehypothecation, such as novation
(see Duffie [2010]), is left for further research. Unsecured lending in a long-term relationship, as in
Kehoe and Levine [1993], is not considered here.
Disruption in the repo market between money market mutual funds (MMMF) and brokerdealers also played a major role in the crisis. I abstract from these issues and focus instead on
the repo market between dealer banks. Gorton and Metrick [2012b] show that during the financial
crisis, MMMF did not reduce net lending to the repo market as a whole. The disruption in the
repo market took place in the bilateral repo market between broker-dealers or hedge funds and
broker-dealers. Unsecured lending though such markets as the Fed funds market is not allowed (see
Afonso and Lagos [2012] for a model about the market). Operational risk such as settlement fails
in the repo market (see Fleming and Garbade [2005]) is also not modeled.
The rest of the paper is organized as follows. Section 2 introduces in more detail institutional
features that the model aims to capture and stylized facts of the crisis that motivate the exercise.
Section 3 introduces the model. In section 4, I formally define the model’s dynamic equilibrium.
Section 5 discusses efficiency of the model. In section 6, I characterize the equilibrium with rollover
of the repo contract, study the efficiency gain of using repo contracts and characterize the efficiency
6
loss from maturity mismatch and collateral risk. Section 7 studies the stability of the shadow
banking system by looking at the transition dynamics triggered by a small but unexpected increase
in collateral risk. Section 8 concludes the paper.
2
Shadow banking and the repo market: institutional features
and stylized facts
In this section, I explain in more detail institutional features and stylized facts of collateral risk,
the repo market, and the shadow banking system.7
Shadow banking and repo lending between financial intermediaries In many financial
markets, either the demand side or the supply side involves mainly agents from the real sector,
whereas in the repo market, many participants are financial intermediaries who could be on either
side of the market. In particular, broker-dealers lend to each other in the repo market as part of the
process of credit creation in the shadow banking system. Hedge funds and broker-dealers implement
arbitrage strategies with each other using securities lending contracts. This trade reduces the cost
of investment and arbitrage for the participants.
Credit creation in the shadow banking system typically starts with loan origination and loan
warehousing and ends with wholesale funding provided by such institutions as MMMF. Before the
loans reach the final cash suppliers, they need to be packaged into asset-backed securities (ABS),
collateralized debt obligations (CDO) and asset-backed commercial paper (ABCP), which typically
involves securities issuance, warehousing, tranching and intermediation.8 The intermediate stages
take time and rely on financing through the interbank lending market. An intermediation chain
comprised of broker-dealers and other intermediaries of the shadow banking system is formed during
the process. This intermediation chain within the shadow banking system relies heavily on shortterm collateralized loans, such as repo. Repo financing in these intermediate steps is not directly
financed by the final cash suppliers, but by other broker-dealers and financial intermediaries. This
market-based intermediation is what distinguishes a shadow banking system from a traditional
7
See, for example, Gorton and Metrick [2012a] and Copeland et al. [2012] for more details on the institutional
features related to the repo market and the financial crisis.
8
See Pozsar, Adrian, Ashcraft, and Boesky [2013].
7
(commercial) banking system.
Maturity mismatch, solvency and liquidity of financial intermediaries Agents in the repo
and securities-lending markets rely on short-term debt to finance investments of longer maturity.9
The maturity mismatch links the liquidity of the short-term lending market to borrowers’ solvency.
In the case of Lehman Brothers, the crisis started from the asset side rather than from short-term
financing per se. The CDOs, – illiquid, long-term investments – that Lehman was initiating started
losing money and became hard to sell long before the crisis. To wait for the investment to turn
around, the bank had to roll over the debt. In the end, Lehman lost the race, ran out of collateral
and was forced into bankruptcy. The bankruptcy was not necessarily a self-fulfilling run. According
to a Wall Street Journal report: “Six weeks before it went bankrupt, Lehman Brothers Holdings
Inc. was effectively out of securities that could be used as collateral to back the short-term loans
it needed to survive.” And Lehman had to rely on “Repo 105”, a way to borrow against collateral
without exposing its high leverage to the public, as early as the end of 2007. Even without a run,
the bank may have had to default as it ran out of collateral.10
Speculation and arbitrage through securities lending also typically involve maturity mismatch.
Convergence trades, which involve going long on one asset and short on a similar asset, typically
involve maturity transformation, as a difference in liquidity is often associated with the spread
between similar assets, and it takes time to realize the arbitrage profit. If it takes longer than
expected to realize those gains from trade, the arbitrageurs may run out of funding and end up
insolvent, as Lehman Brothers did. The downfall of Long-Term Capital Management11 and MF
Global12 are similar cases.
In all these cases, the solvency of a financial intermediary depends on the maturity of investment
projects and the liquidity of the repo or securities lending market. The solvency and liquidity of a
financial intermediary is best understood in terms of an equilibrium model.
9
10
Adrian et al. [2009] shows that about 65% of outstanding repos of primary dealers are overnight repos.
“Repos Played a Key Role in Lehman’s Demise”,
http://online.wsj.com/article/SB10001424052748703447104575118150651790066.html
11
http://en.wikipedia.org/wiki/Long-Term_Capital_Management#Downturn
12
http://en.wikipedia.org/wiki/MF_Global
8
Figure
1:
Distribution
of
home
price
changes
by
county
(from
NYFed
website:
http://www.newyorkfed.org/home-price-index/)
Collateral risk in the 2007-2008 financial crisis
According to the home-price-index of the New York Fed,13 the growth rate of housing prices slowed
down before 2007 and turned negative close to the end of 2007. Figure 1 illustrates the year-overyear changes in housing prices in the US at the county level. In August 2007, 50% of counties
experienced negative price changes and at the end of 2007, more than 75% of counties started to
show a decline in housing prices. The grey area in the figure marks the great recession.
As housing prices spiraled downward, the riskiness of mortgage-backed securities (MBS) increased. Figure 2
14
shows the market price index for a credit default swap contract that provided
insurance against the default risk of a pool of mortgage-backed securities issued in early 2006. The
discrepancy between the par value, 100, and the actual price index measures the shift in market beliefs about the riskiness of the mortgage-backed securities. The figure shows the index for tranches
with a AAA rating to tranches with a BBB- rating. Before mid-2007, the market’s belief about the
13
14
http://www.newyorkfed.org/home-price-index/
According to http://www.nera.com/nera-files/PUB_Subprime_Series_Part_IX_0412.pdf.
9
Figure 2: ABX index and risk of mortgage backed securities.
riskiness of all tranches barely moved. Riskiness of tranches with lower ratings increased first, at
the beginning of 2007, and then in July 2007, riskiness of AAA tranches increased from zero to a
positive number and kept increasing. The sudden changes in the price index imply that the shift
in market beliefs came as an unexpected shock.
Why was there a sudden shift in the market belief about collateral risk? Dang et al. [2009,
2012] relate this to the information sensitivity of the debt contract and the lemons problem of
MBS. However, the link between the changes in housing prices and the riskiness of mortgagebacked securities indicates that the risk is more likely to be related to the unexpected collapse
of the housing market in the whole country, rather than to the lemons problem that arises from
the quality deterioration of a fraction of mortgages. And the collapse of the housing market is so
widely publicized that asymmetric information on this fact was unlikely. Thus in this paper, I take
the view that the crisis was triggered by an unexpected increase in collateral risk, rather than by
market failure due to a lemons problem.
Balance sheet adjustments of financial intermediaries and liquidity of the bilateral
repo market
He et al. [2010] estimates that, on the asset side, hedge funds and broker-dealers
10
reduced holdings of securitized assets by approximately $800 billion during the 2008 crisis. It was
not just the size of broker-dealers’ balance sheets that changed; the composition changed as well.
Before the crisis, in November 2007, credit and mortgage-related assets made up of 32% of the
total value of the trading assets of Goldman Sachs, Morgan Stanley and Merrill Lynch. After the
crisis, in March 2009, they accounted for 23%.15 The flight to such safe assets as treasury bills
may be related to the increasing market risk of securities and is also consistent with their reduced
activity in the repo market, where the haircut for risky collateral assets increased sharply.16 Gorton
and Metrick [2012b] finds through Flow of Funds data that both the Repo assets and liability of
broker-dealers shrank during the crisis, indicating the freeze of the repo market and broker-dealers’
reduced activity in the market, as illustrated in Figure 3. This evidence implies that not only
financial intermediaries’ balance sheet, but also the portfolio composition of the balance sheet in
the shadow banking system, may have contributed to and been affected by the financial crisis.
Krishnamurthy et al. [2012] shows that funding from cash providers such as MMMF did not
change dramatically during the crisis. This implies that the dramatic change happened in the
bilateral repo market between broker-dealers and hedge funds, which is where increasing haircut
is reported in Gorton and Metrick [2012a,b] and Hördahl and King [2008]. This is consistent
with Gorton and Metrick [2012b]’s finding that repo lending of investors other than money market
mutual funds shrunk dramatically during the crisis. (See Figure 3 for dynamics in repo lending
and borrowing. See Figure 4 for dynamics in haircut.)
3
The model
The model is set in continuous time. The economy starts at t = 0 and lasts forever. There is a
continuum of agents of constant measure. At any moment, there is a constant inflow of entrants,
÷, and an equal outflow of exits.
There is a durable consumption good in the economy and there are some productive trees. A
15
See Table 7 of He et al. [2010]. The definition of a trading asset is “a collection of securities held by a firm
that are held for the purpose of reselling for a profit. Trading assets are recorded as a separate account from the
investment portfolio.” (http://www.investopedia.com/terms/t/trading-assets.asp)
16
For example, Gorton and Metrick [2012a] documents the devaluation of BBB asset-backed securities and a sharp
increase in haircut in the bilateral repo market during the crisis.
11
Broker-dealers’ repo assets and liabilities, in $billions
Figure 3: The freeze of the repo market. (from Gorton and Metrick [2012b])
Figure 4: Collateral quality and haircut dynamics during the crisis. (from Gorton and Metrick
[2009])
12
tree bears consumption goods only at its maturity date. The maturity date of a tree is random
and follows an idiosyncratic Poisson process.
Agents are ex ante homogeneous. They are endowed with a0 œ R++ units of collateral trees
and m0 œ R++ units of consumption good when they enter the economy.
´T
An agent’s expected payoff at time t is Et t cu e≠fl(u≠t) du, where cu du is the measure of apples
she consumes between u and u + du, fl is the discount factor and T is the random moment when
she leaves the economy.
A collateral tree matures with Poisson rate µ œ R++ . If a collateral tree matures at date t,
it bears y · Êt apples at maturity date t, with y œ R++ . Êt is the aggregate state of the economy
at date t; it represents the aggregate risk that the quality of a collateral asset may deteriorate, or
the aggregate collateral risk. There are two aggregate states, Êt œ {0, 1}. When Êt = 1, every tree
bears y apples if it matures at t, and when Êt = 0, every tree bears no apples. I assume that the
economy is in the good state initially, Ê0 = 1. And the bad state, Ê = 0, is assumed to be an
absorbing state. The arrival of the bad state follows a Poisson process with rate ‰ œ R++ . As the
likelihood of shock, Êt = 0, can be small or large, the model applies to collateral assets of both
high and low quality.
Another type of tree in the economy represents investment opportunities. An investment opportunity is a long-term technology that transforms consumption goods at the investment date into
consumption goods at the (random) maturity date. The maturity date arrives with Poisson rate
fi œ R++ . The investment is one-shot and doesn’t require additional resources at subsequent dates.
If an agent does not exploit the investment opportunity the moment it arrives, she loses it. With
i units of consumption good as input at the investment date, the output at maturity is
fl+fi
fi f (i),
where f : R++ æ R++ . In the benchmark model, I assume that the production function takes the
form, f (i) = ◊i– , with productivity parameter ◊ œ R++ and – œ (0, 1), so f (i) is concave and the
marginal output at zero input is infinity. I assume that only the agent who invests in a long-term
technology has the skill to manage it. The output from the technology if other agents own the
project is zero, and other agents cannot take the output from it away from the investor.
The long-term investment opportunity is endowed to agents with delay, which represents the
search friction to find a profitable investment opportunity. After entering the economy, each agent
receives at most one investment opportunity, at a random date. The arrival date of an agent’s
13
Figure 5: The life cycle of an agent in Autarky.
investment opportunity follows an idiosyncratic Poisson process, arrival rate ⁄ œ R++ .
An agent leaves the market after her collateral asset and the long-term project mature. I assume
for simplicity that agents lose their chance to find an investment opportunity after their collateral
matures.
Figure 5 describes a realization of the life cycle of an agent in Autarky. Upon entry, the agent
decides her consumption and storage. She stores some of the consumption good in order to provide
for investment in the long-term technology. When the investment opportunity arrives, she draws
her consumption good from storage to invest in the project and then waits for the project to mature.
She leaves the market after both her asset and the project mature.
Since storing consumption is not productive and delays consumption, a financial system can
improve efficiency by allocating consumption goods in storage to agents who need more funding to
take advantage of the investment opportunity.
The repo market allows agents waiting for long-term projects to earn interest by lending, and
agents with long-term projects can increase their investment by borrowing against their collateral.
The repo market is assumed to be perfectly competitive. A repo contract has three components:
the interest rate Rt œ R+ , haircut ht œ [0, Œ) and maturity dt. According to the contract, a
borrower puts down ht units of collateral for each unit of consumption she borrows from the lender
at the moment of signing the contract. At maturity, date t+dt, if no party defaults on the contract,
14
then the borrower pays 1 + Rt dt units of consumption good for each unit she borrows at t, and the
lender delivers ht units of collateral asset back to the borrower for each unit of borrowing.17 Given
the contract, an agent with asset holding a can borrow up to
a
ht .
At the maturity date of a repo contract, the borrower can choose to default or repay the debt.
There are two ways to repay the debt: by using consumption goods that the borrower stored from
her endowment or that she obtained when her tree matures; or by borrowing from other lenders.
This is what I refer to as repo rollover. As long as a borrower has not borrowed up to
a
ht ,
she can
choose to roll over her debt.
For lenders, as in Acharya and Bisin [2013], I assume that the repo market is an opaque, overthe-counter market, so the repo contract is not conditional on information such as the borrowers’
balance sheet.18 At any moment, one repo contract clears the whole market. I assume that every
lender is assigned to one borrower in the market-clearing process.19 After the repo market clears,
lenders meet their counterparty. At that moment, lenders can see borrowers’ information and
whether or not they will default. Then, lenders can decide whether to carry out the repo contract
or to reject the borrower and wait until next period.
Since a lender is matched with one borrower, the counterparty risk, the risk that a borrower may
default, is undiversified. Denote the probability that the counterparty defaults on a repo contract
signed at t as ”t dt. Counterparty default is a shock to the lender’s portfolio. For a lender with a
units of asset and s units of apple in repo account, she will hold a + ht s units of asset but will have
no funding left in the repo account when default happens. If default happens with probability one,
the contract is observationally equivalent to a spot transaction in which the borrower sells the asset
to the lender, at unit price
1
ht .
To simplify the analysis, I impose the following restrictions on agents’ strategies. A borrower
is not allowed to borrow or lend in the repo market after she defaults. A lender is not allowed
to move additional funding from storage to the repo market after she loses all her funding to the
17
Default does not incur any loss to the defaulting agent other than the collateral asset in the repo contract. For
example, for an agent with a long-term investment, default does not affect her return from the investment because
the investment generates only private return for the borrower that no one else can control.
18
Trading delay due to search friction, as inDuffie et al. [2005], Lagos and Rocheteau [2009], Afonso and Lagos
[2012], is also ignored here.
19
A lender can only lend to one borrower, a borrower can borrow from several lenders.
15
repo market when she meets a defaulting borrower or she herself defaults on a repo contract.20
With these two simplifications, we focus on the borrowing decision of agents who need to finance
their long-term investment and lending decisions of those lenders who have not met a defaulting
borrower. Additionally, I assume that interest payments from repo lending must be consumed and
cannot be accumulated for repo lending or storage.
3.1
Discussion
If all borrowers default with probability one, the market is equivalent to a market for trading the
asset; however, agents cannot choose to buy or sell the asset and lend or borrow against the asset.
What is missing is an additional market for a firesale asset. In such a case, lenders do not worry
about counterparty default risk, and buyers of the asset can optimize their portfolio based on their
risk exposure. While the current setup is in line with the observation that the repo market is much
more liquid than the market for trading assets, I will study the effect of introducing an additional
market in the extension.
I also assume that the default over the collateral does not affect agents’ payoff from their project
investment. Default does not lead to bankruptcy, and collateral delivery is the only requirement if
a borrower defaults. The separation of default from bankruptcy allows the agent to buy and sell
the collateral asset through repo contracts (and default). And this is consistent with the exemption
of repo contracts from “automatic stay” (Garbade [2006]). The separation also allows the model
to characterize credit derivative markets. I also assume that default does not incur loss to the
borrower other than the collateral she puts down because other investors cannot separate he from
her long-term investment. This is consistent with moral-hazard frictions that lenders may face.
Another property of the repo contract is that dividends from the collateral asset before the
maturity of the repo contract belong to the lender. This is not an issue for overnight repo, as no
dividend is generated from an asset overnight. Likewise, since the maturity of a repo contract is
assumed to be infinitesimal in my model, there will not be any dividend payment from the asset
before the maturity of the contract.
I assume that lenders face undiversified counterparty risk. Although banks in the real world
20
When losing her funding of amount s to the repo market, she has s(1 + h) additional collateral asset. But she
can still borrow from the repo market.
16
are large enough to diversify the idiosyncratic coutnerpary risk in normal times, diversification
would be impossible when default is triggered by certain aggregate shocks. My assumption will
be more useful when studying the dynamic response of the financial system to aggregate shocks.
Another reason I make this assumption is that I am modeling the repo market between financial
intermediaries. As financial intermediaries borrow and lend on a large scale and the total number of
financial intermediaries is limited, undiversified counterparty risk is more relevant for the inter-bank
repo market.
4
Equilibrium definition
In this section, I give the formal definition of the dynamic equilibrium with a certain initial distribution of agents and initial aggregate state. In the equilibrium definition subsection, I first formalize
the individual agent’s problem, after which I define the law of motion of the economy. And in the
end, I define the equilibrium.
4.1
An agent’s problem
An agent’s problem depends on whether or not the repo market functions, which, in turn, depends
on the aggregate state.
When Êt = 0, the repo market is not functioning, as the supply of valuable collateral asset in
this state is 0.
When Êt = 1, the repo market is functioning. Then, an agent’s problem depends on her portfolio, her lending/borrowing history and whether she is still looking for an investment opportunity or
she has already invested in a project. In that case, the life cycle of an agent who manages to find a
long-term project is illustrated in Figure 6. After entry, the agent allocates her consumption good
endowment to consumption, storage and repo lending. Before she finds her long-term technology,
she continues making decisions on consumption, storage and repo lending.21 When she finds her
long-term project, she withdraws consumption goods from storage and repo lending, borrows from
the repo market and invests in the long-term project. Before her long-term investment matures,
she does not have consumption goods left so she decides whether to roll over her debt or default. If
21
When lending to other agents, the agent may meet a default borrower with Poisson rate ”. If that happens, her
lending turns into additional asset holdings and she loses her capacity to continue lending.
17
Figure 6: life cycle of an agent with access to the repo market
her long-term investment or asset matures when she is still rolling over her borrowing, she repays
her outstanding debt, consumes the remaining consumption and waits for the rest of her trees to
mature. After all her asset matures and she consumes all the consumption she owns, she leaves the
market.
To solve an agent’s problem when Êt = 1, I go backward. I first solve her problem after project
investment; then, I solve her problem at long-term investment, her problem before the investment,
and, finally, her problem at the beginning of her life. I then explain an investor’s problem when
Êt = 0. Table 1 summarizes the value function and policy functions for an agent’s problem when
she faces different situations. I now explain an agent’s problem in different situations.
Situation 1: Êt = 1, agent’s problem after LT investment
Wt (a, m, s, i) = max
c + EWt (a, mÕ , sÕ , i),
Õ Õ
c,m ,s
s.t.
(1)
c + mÕ + sÕ Æ s + m,
sÕ Ø ≠ hat ,
c, mÕ Ø 0,
where the first constraint is the agent’s budget constraint; the second is the collateral constraint,
18
situation
value function
policy function
distribution
consumption c̃1t
storage m̃1t
Êt = 1, after LT investment
Wt (a, m, s, i)
net lending s̃1t
F1t (a, m, s, i)
acceptance z̃1t ,
default d˜1t
consumption c̃It
Êt = 1, at LT investment
Ut (a, m, s)
storage m̃It
net lending s̃It
0
investment ĩIt
consumption c̃0t
storage m̃0t
Êt = 1, active before LT investment
Vt (a, m, s)
net lending s̃0t
F0t (a, m, s)
acceptancez̃0t
default d˜0t
Êt = 1, deactivated before LT investment
Vtd (a, m)
Êt = 0, before LT investment
VtA (a, m)
consumption c̃dt
storage m̃dt
consumption c̃A
storage m̃A
Table 1: Value functions and policy functions
19
Fdt (a, m)
G0t (a, m)
which imposes that borrowing cannot exceed
as
1+h ;
and the last two are non-negativity constraints
for consumption and storage allocation. Agents choose consumption c, storage mÕ and repo lending
sÕ to maximize the payoff from consumption and continuation value EWt (a, mÕ , sÕ , i), which depends
on random events between t and t + dt.
EWt (a, m, s, i)
5
(2)
6
=µdt a + m + s + f (i) + fidt
5
5
6
5
6
+”t dtIs>0 e≠fldt max zWt+dt (a + ht s, m, 0, i) + (1 ≠ z)Wt+dt (a, m + s, 0, i)
zœ[0,1]
6
µ
fl+fi
a+m+s+
f (i) + ‰dt m + f (i)
fl+µ+‰
fi
+ (1 ≠ (µ + fi + ‰ + ”t dtIs>0 )dt) e
≠fldt
5 3
4
µy
max d
(a + ht s) + m + f (i)
fl+µ+‰
dœ[0,1]
6
+ (1 ≠ d) (Rt dts + Wt+dt (a, m, s, i)) Is>0 + (1 ≠ d)Wt+dt (a, m, (1 + Rt dt)s, i)IsÆ0 .
The continuation value depends on several random events. With probability µdt, the agent’s asset
matures. In this case, she repays ≠s apples to her lenders, if s < 0, or withdraws s apples from
inter-bank lending, consumes a+m+s apples, and waits for her long-term project to mature, which
delivers expected payoff f (i). With probability fidt, the agent’s long-term investment matures. In
this case, she repays her debt or draws lending from the repo market, consumes m + s +
apples, and waits for her collateral asset holdings to mature, which delivers expected payoff
fl+fi
fi f (i)
µ
fl+µ+‰ a.
With probability ‰dt, the aggregate devaluation shock hits the economy and the collateral asset
becomes worthless. So if s < 0, she default on her debt, consumes m apples that she withdraws
from her storage account and waits from her long-term investment to mature. If s > 0, on seeing
the aggregate shock hitting the economy, she withdraws her apples from the repo market and
consumes the s apples immediately. The next term is the expected payoff from the event that the
counterparty defaults. This term shows up only if the agent is lending in the repo market. So, if
s > 0, then with probability ”t dt, the agent meets a defaulting borrower. In this case, her payoff
depends on whether she is willing to lend to the borrower after knowing that the borrowing is going
to default. If she decides to lend to the defaulting borrower, her loan s will turn into ht s units of
additional asset holdings when the repo contract matures in dt period. Thus, her continuation value
with this choice is e≠fldt W (a + ht s, m, 0, i). If she decides not to lend to the defaulting borrower,
she adds s additional apples to storage account for dt period. So her continuation value with this
choice is e≠fldt W (a, m + s, 0, i). With the residual probability, [1 ≠ (µ + fi + ‰ + ”t dtIs>0 )dt], the
20
agent meets a non-defaulting borrower and decides whether or not to default. If she defaults, her
continuation value is e≠fldt
Ë
µy(a+ht s)
fl+µ+‰
È
+ m + f (i) , where the first component is the expected payoff
from the asset holdings, m is the payoff from the consumption of apples withdrawn from the storage
account and f (i) is the expected payoff from the long-term investment. If she does not default, she
will get Rt dts apples as interest payment, so her continuation value is e≠fldt W (a, m, (1 + Rt dt)s, i).
Refraining from lending is ruled out here as long as interest rate Rt is positive. If s < 0, she needs
to borrow from the repo market. Similarly, e≠fldt
Ó
µy[a+ht s]
fl+µ+‰
Ô
+ m + f (i) is her payoff from default,
and e≠fldt W (a, m, (1 + Rt dt)s, i) is her payoff from rolling over her debt.
Debt rollover is an option only if (1 + Rt dt)s Ø ≠ hat . So, if the state variable s = ≠ hat , the
agent’s borrowing constraint is already binding, so she must default. Thus,
3
EWt a, m, ≠
4
a
, i = m + f (i),
ht
(3)
which is equal to m + f (i) plus some term of the same order of magnitude as the infinitesimal
period dt.
Situation 2: Êt = 1, agent’s problem at long-term investment
Ut (a, m, s) = max
c + Wt (a, mÕ , sÕ , i)
Õ Õ
c,m ,s ,i
s.t.
(4)
c + sÕ + mÕ + i Æ s + m,
sÕ Ø ≠ hat ,
c, mÕ , i Ø 0,
where agents choose consumption, c, storage, mÕ , lending or borrowing through the repo market,
sÕ , and investment in the long-term project, i, to maximize their payoff from consumption and continuation value Wt (a, mÕ , sÕ , i). The constraints the agent faces are resource constraint, borrowing
constraint and non-negativity constraints of the choice variables, c, mÕ and i.
21
Situation 3: Êt = 1, agent’s problem before long-term investment
Vt (a, m, s) =
s.t.
max
c,sÕ ,mÕ ,z,d
c + EVt (a, mÕ , sÕ ),
(5)
c + sÕ + mÕ Æ s + m,
sÕ Ø ≠ hat ,
c, mÕ Ø 0,
which is similar to agents’ problem after project investment, equation 1.
EVt (a, m, s)
(6)
=µdt (a + m + s) + ⁄dtUt (a, m, s) + ‰dtVtA (m)
5
6
d
+”t Is>0 dte≠fldt max zVt+dt
(a + ht s, m) + (1 ≠ z)Vt+dt (a, m, s)
zœ[0,1]
+ [1 ≠ (µ + ⁄ + ‰ + ”t Is>0 )dt] max e
dœ[0,1]
≠fldt
5
dVt+dt (a + ht s, m, 0)
6
+ (1 ≠ d) (Rt dts + Vt+dt (a, m, s)) Is>0 + (1 ≠ d)Vt+dt (a, m, (1 + Rt dt)s)IsÆ0 .
The continuation value has a similar expression to the continuation value 2. The additional random
event is the event of finding a long-term project. With probability ⁄dt, the agent finds a longterm project in dt period. The continuation value contingent on the event is Ut (a, m, s). With
probability µdt, the agent’s asset matures.22 And if a lender lends to a defaulting borrower, she
d (a + h s, m).
will be deactivated, with continuation value, Vt+dt
t
If s = ≠ hat , the agent’s borrowing constraint is already binding, then she must default. So,
3
EVt a, m, ≠
a
ht
4
= VtA (m).
(7)
After default, the agent has no asset left. So her continuation value is the same as what she would
have been under Autarky with storage m.
The problem of an agent deactivated because of lending to a defaulting borrower is
22
The collateral asset of an agent is assumed to mature at the same time. This is not exactly consistent with the
assumption that the maturity of collateral asset is idiosyncratic across agents because the asset an agent receives
from other agents is assumed to mature at the same time as his own asset.
22
Vtd (a, m) = maxÕ c + EVtd (a, mÕ ),
c,m
s.t.
where the continuation value is,
(8)
c + mÕ Æ m,
c, mÕ Ø 0,
EVtd (a, m) = µdt (a + m) + ⁄dtUt (a, m, 0) + ‰dtVtA (m) + [1 ≠ (µ + ⁄ + ‰)dt] e≠fldt Vtd (a, m). (9)
The agent finds her long-term project with probability ⁄dt and her continuation value is Ut (a, m, 0).
Situation 4: Êt = 0
The value function, VA (m) and policy functions, c̃A (m), m̃A (m), solve the problem of an agent’s
problem before she finds her long-term projects.
VA (m) = maxÕ c + EVA (mÕ ),
c,m
s.t.
(10)
c + mÕ Æ m,
c, mÕ Ø 0,
where the agents choose consumption c and storage for the next period mÕ to maximize their payoff
from consumption and continuation value E ṼA (mÕ ), subject to the resource constraint and nonnegativity constraints on consumption and storage. The continuation value depends on the random
events that may happen during dt period.
EVA (m) = ⁄dt max {m ≠ i + f (i)} + µdtm + [1 ≠ (⁄ + µ)dt] e≠fldt V A (m).
0ÆiÆm
With probability ⁄dt, the agent finds her long-term investment and, in that case, chooses optimally
to invest in the long-term project and consumption so as to maximize her expected payoff c + f (i),
where c = m ≠ i, subject to non-negativity constraints of consumption and investment. With
probability µdt, her collateral asset matures and bears no apples at maturity and she loses her
chance to find a long-term investment opportunity in the future. In that case, she consumes away
the apples in the storage account and leaves the market. With the residual probability 1≠(⁄+µ)dt,
nothing happens during the dt period so her continuation value is e≠fldt V A (m).
23
Figure 7: Law of motion of an agent’s state before the realization of collateral shock (Êt = 1).
Laws of motion for the distribution of agents
The exact law of motion is left to the Appendix. Here, I illustrate the laws of motion on the
equilibrium path, where Êt = 1, ’t using Figure 7. All new entrants enter as active lenders and
are counted in the distribution F0t . A lender leaves the pool of active lenders in three situations.
If her collateral asset matures before she finds her long-term project, she leaves the economy after
consuming all her consumption goods. If she meets a defaulting borrower, she enters the pool
of deactivated lenders with distribution Fdt . If she finds her investment opportunity, she enters
the pool of borrowers with distribution F1t . An agent in the pool of deactivated lenders leaves the
economy if her collateral asset matures before she finds a project, or she enters the pool of borrowers
when she finds her long-term project. An agent in the pool of borrowers exits the economy after
both her project and collateral asset mature. She will stop borrowing when her collateral asset or
project matures, or when she defaults on her loan.
Equilibrium definition
Definition 4.1. An equilibrium with initial distribution F10 , Fd0 and F00 and initial state Ê0 = 1,
is the repo contract term {Rt , ht }tØ0 , default rate, {”t }tØ0 , agents’ policy functions and value
functions and aggregate law of motion such that,
24
(i) given R, h and ⁄, agents’ policy functions and value function solve their problems if Êt = 1.
(ii) agents’ policy functions and value functions solve their problem if Êt = 0.
(iii) the contract (R, h) clears the repo market given the distribution of agents and agents’
decision function,
ˆ
s̃1t (a, m, s, i)dF1t (a, m, s, i) +
ˆ
s̃0t (a, m, s)dF0t (a, m, s) = 0, if Êt = 1, ’t.
(iv) the distribution of agents is endogenously determined by laws of motion.
(v) agents’ expectation on default rate ”t is consistent with the actual default rate,
´
´
1
1
˜
˜
{(a,m,s):sÆ0} d1t (a, m, s, i)sdF1t (a, m, s, i) + {(a,m,s):sÆ0} d0t (a, m, s)sdF0t (a, m, s)
´
´
”t dt =
, for Êt = 1.
1
1
{(a,m,s):sÆ0} sdF1t (a, m, s, i) + {(a,m,s):sÆ0} sdF0t (a, m, s)
”t dt is the probability that a lender’s counterparty may default on her borrowing. Since the
lender is ignorant of the borrower’s portfolio, the lender’s counterparty can be thought of as a
random draw from the pool of borrowers, weighted by borrowers’ funding demand. Suppose that
an agent borrows to invest in a long-term project, and she also has invested all her apples in
storage and in interbank lending. When the initial borrowing matures in dt period, the cumulative
probability that her project or collateral asset matures is (fi + µ)dt, which is negligible. She has two
choices, to roll over her debt or to default. If every borrower chooses to default immediately after
initial borrowing, ”t dt = 1. ”t dt < 1 when some agents choose to rollover their debt in equilibrium.
Definition 4.2. A stationary equilibrium with initial state Ê0 = 1 is an equilibrium with initial
distributions F10 , Fd0 and F00 and initial state Ê0 = 1, such that F0t (a, m, s) = F00 (a, m, s),
Fdt (a, m) = Fd0 (a, m), and F1t (a, m, s, i) = F10 (a, m, s, i), ’t.
5
Efficiency
To understand the welfare loss from the market incompleteness, I study properties of efficient
allocation in this section.
The social planner’s choice is aggregate consumption flow, C· , for each newly born agent at
moment · , aggregate storage, m· , and investment in long-term tech, i· , for each agent with an
opportunity to invest in the long-term technology.
The social planner’s problem at moment t is
25
St =
max
{C· ,m· ,i· }’· Øt
ˆ
Œ
C· e≠fl(· ≠t) d·,
t
÷
C· d· + (⁄d· ) µ+⁄
i· + m·
s.t.
C· , m· , i·
fl+fi –
≠fi(· ≠s) ⁄ ÷ ds
µ+⁄
≠Œ fi ◊is (fid· ) e
´·
≠µ(·
≠s)
+ ≠Œ ayÊ· (µd· ) e
÷ds + (÷d· ) M·
Æ
´·
Ø 0.
+ m· ≠d· ,
In the objective function of the planner, ÷dt is the measure of agents born during · and · +
dt, and e≠fl(· ≠t) is the discount factor for future welfare gain. The planner chooses allocation,
{C· , m· , i· }’· Øt , to maximize the objective function, subject to nonnegative constraints of the
choice variables and the resource constraint. The right-hand side of the resource constraint includes
apples from maturing long-term projects from past investments, maturing collateral from agents
born before · , and production from short-term technology and storage technology.
fl+fi –
fi ◊is
is the
measure of fruits from a maturing project created at moment s with investment is . (fid· ) e≠fi(· ≠s)
÷
is the probability that a project created at moment s matures between · and · + d· . And ⁄ µ+⁄
ds
is the measure of projects created between moment s and s + ds. Similarly, ayÊ· is the measure
of fruits from the maturing asset of an agent. (µd· ) e≠µ(· ≠s) is the probability that the asset of an
agent born at moment s matures between · and · + d· . And ÷ds is the measure of agents born
between s and s + ds. The left-hand side of the resource constraint is the allocation of the available
÷
apples to consumption, long-term investment and storage. (⁄d· ) µ+⁄
is the measure of projects
found between · and · + d· .
Proposition 5.1. The efficient allocation {C· , m· , i· }’· Øt is
i· = iú , such that ◊–iú–≠1 = 1,
m· = 0.
And the efficiency of the economy is characterized solely by investment allocation iú
5
6
fl + fi ú–
⁄÷
C· =
◊i ≠ iú
+ ayÊ· ÷.
fi
µ+⁄
As a benchmark for comparison, in Autarkic allocation, where the repo market is shut down,
individual investment, i, is constrained to be equal to the amount of apples agents are endowed with
and store using the storage technology until they find the project. If the initial endowment is small
26
enough so that the marginal return from storing all endowments is greater than 1,
the expected return from investment is
marginal return from storage is equal to
in long-term technology is greater than
⁄◊m–
0 +µm0
⁄◊–m–≠1
+µ
0
fl+µ+⁄
> 1,
fl+µ+⁄ . Otherwise, the storage is such that the expected
⁄◊–m–≠1
+µ
0
1, fl+µ+⁄
= 1. So, the marginal return from investment
fl+⁄
⁄ . Therefore, there exists a wedge between the marginal
return of the project and the marginal utility of consumption in Autarkic allocation, which is
greater than
fl+⁄
⁄
≠ 1 = ⁄fl . The lower bound of the wedge increases with the search friction to find
the long-term project.
In the efficient allocation, in contrast, the wedge between the marginal return of the project
investment and the marginal utility of consumption is 0. The social planner, not subject to enforceability constraints, can allocate consumption goods from maturity projects to new investment.
Thus, the optimal investment allocation does not depend on the aggregate risk on the value of
collateral. The efficient investment in the long-term project tree does not depend on the value of
the collateral asset. As we will see, the efficiency gain from using repo contracts in this environment
comes from the transfer of output from maturity projects to investment in new projects.
Another difference between Autarkic allocation and the first-best allocation is the allocation to
storage, which can be interpreted as liquidity hoarding. While the storage of each agent in Autarky
is equal to her investment in long-term technology, the aggregate storage is 0 in efficient allocation.
In other words, it is socially wasteful for agents to hoard liquidity when the social planner is not
constrained by market incompleteness. The wedge and inefficiency in the Autarkic economy also
comes from the fact that the return on storage technology is low, so it is costly for agents to use
the storage technology. If the return on storage technology is equal to the discount factor, the
lower bound of the wedge in Autarky would be zero. This is the case, for example, if the storage
technology is a fiat currency and the monetary authority follows Friedman rule.
6
23
Equilibrium characterization
With collateral risk and the state-contingent repo contract, default does not happen in equilibrium
as long as not all agents default. Next, I add maturity mismatch to the analysis. In the full
equilibrium, I will show that maturity mismatch results in repo rollover and equilibrium default.
23
See Lagos and Wright [2005], Berentsen et al. [2007] and Williamson and Wright [2010].
27
I will also show that default triggers more default in equilibrium because of the undiversified
counterparty risk.
6.1
Characterization of an equilibrium with repo rollover
With a short-term repo contract not contingent on liquidity arrival, agents borrow from the repo
market when they find their long-term investment opportunity, and then they roll over their debt
to wait for their long-term investment or collateral asset to mature. But as long as the interest rate
R is positive, they cannot roll over their debt infinitely. At certain point, they will reach their debt
limit, ha . The dynamics of debt holding in the equilibrium with debt rollover is illustrated in Figure
8(a). Liquidity arrives when the borrower’s collateral asset or project matures, which is a random
date that could be earlier or later than the moment of reaching the debt limit. The borrower will
keep rolling over her debt until she repays is using consumption goods from her trees at maturity
or she defaults when she reaches her debt limit.
In this equilibrium, there is a trade-off between default and investment in long-term technology.
This trade-off is illustrated in Figure 8(b). As initial borrowing bÕ increases, the moment of reaching
the debt limit moves to the left of the timeline. Thus, the probability that the borrower reaches
her debt limit before she receives liquidity from maturing trees decreases. If the agent takes on
more initial debt, she gains more from long-term investment but is less likely to receive apples from
long-term investment or collateral asset before she defaults when she reaches her debt limit.
(a) debt rollover
(b) tradeoff for initial borrowing
Figure 8: debt rollover
28
The dynamics of other choice variables in the equilibrium with stationary distribution are as
follows. Before an agent finds her long-term investment, she keeps lending a constant amount
of apples to other agents through the repo market and stores a constant amount of apples. She
consumes all interest payments from lending. If she finds a long-term technology, she invests all
her apples in storage and repo lending and from her initial borrowing to the long-term technology
and stops consuming until the project or her assets mature. If she meets a defaulting agent before
her investment, all her lending turns into assets and her consumption drops to zero.
Necessary conditions for an equilibrium with debt rollover
Lemma 6.1. A necessary and sufficient condition for a borrower with debt holding b to roll over
her debt until reaching her debt limit is
µ
fl+µ+‰ yh
Ø
µ+fi
fl+‰+µ+fi≠R
and R < fl + ‰ + µ + fi.
Through debt rollover, a borrower can avoid losing collateral assets to lenders. The benefit
of avoiding default is higher when haircut is high enough. Lemma 6.1 gives the necessary and
sufficient condition for borrowers to roll over their debt holdings.
For equilibrium rollover to take place, a lender must be willing to lend to defaulting borrowers,
and she does not default on repo lending to non-defaulting borrowers. This means that when she
meets a defaulting borrower, her continuation value from lending to that borrower, e≠fldt V d (a +
hs, m), must be no less than her continuation value from waiting for the next lending opportunity,
e≠fldt V (a, m, s). And when she meets a non-defaulting borrower, her continuation value from
waiting for the debt repayment, e≠fldt V (a, m, s(1 + Rdt)), must be no less than her continuation
value from defaulting e≠fldt V d (a + hs, m).
Lemma 6.2. A sufficient and necessary condition for lenders’ strategy in the equilibrium with debt
rollover to be optimal is: V (a, m, s) = V d (a + hs, m).
Lemma 6.2 implies that
µ (yh ≠ 1) s = sR + ⁄ [U (a, m, s) ≠ U (a + hs, m, 0)] ,
which shows that haircut compensates for two losses from lending to a defaulting borrower instead
of a non-defaulting borrower: the loss of interest payment; and the difference in the continuation
29
value when she invests in long-term technology. When the arrival rate of the aggregate shock
increases, the heterogeneity increases and haircut may be more likely to satisfy the condition for
borrowers to roll over their debt in Lemma 6.1.
Equilibrium characterization
Lemma 6.3. If the repo contract satisfies the condition in Lemma 6.1, the value function W (a, 0, ≠b, i)
for b œ R++ in the equilibrium with debt rollover is characterized by the following differential equation
flW (a, 0, ≠b, i) = fi
5
5
6
fl+fi
µ
f (i) +
ya ≠ b ≠ W (a, 0, ≠b, i)
fi
fl+µ+‰
6
5
(11)
6
+ ‰ f (i) ≠ W (a, 0, ≠b, i) + µ f (i) + ya ≠ b ≠ W (a, 0, ≠b, i)
+
ˆW (a, 0, ≠b, i)
bR
ˆb
and boundary condition
3
4
a
W a, 0, ≠ , i = f (i).
h
(12)
The solution to the differential equation is
5
6 3
µ+fi
1
µ
hb
W (a, 0, ≠b, i) =
≠
y a
fl+‰+µ+fi≠Rh fl+µ+‰
a
µ+fi
µ
≠
b + f (i) +
ya
fl+‰+µ+fi≠R
fl+µ+‰
4(fl+‰+µ+fi)/R
.
(13)
The initial borrowing at the moment the agent receives an investment opportunity is pinned
down by problem (4). The trade-off between losing assets through default and more investment in
the long-term project leads to the following result about initial borrowing.
Lemma 6.4. If h >
µ+fi
R ,
µ+fi
fl+µ+‰
fl+‰+µ+fi≠R µy ,
f Õ (s+m) >
µ+fi
fl+‰+µ+fi≠R
and f Õ (s+m+ ha ) <
µ fl+‰+µ+fi
R fl+µ+‰ yh≠
an investor’s initial borrowing, b, is solved by
fl+‰+µ+fi
R
3
hb
a
4(fl+‰+µ+fi≠R)/R
Initial borrowing b = 0 if f (s+m) Æ
µ+fi
fl+‰+µ+fi≠R
=
f Õ (s + m + b) ≠
and b =
30
µ
fl+µ+‰ yh
a
1+h
≠
µ+fi
fl+‰+µ+fi≠R
µ+fi
fl+‰+µ+fi≠R
if f Õ (s+m+ ha ) Ø
.
(14)
µ fl+‰+µ+fi
µ+fi
R fl+µ+‰ yh≠ R .
The first condition for Lemma 6.4 is the sufficient and necessary condition for debt rollover
stated in Lemma 6.1. If the marginal return from long-term investment when an agent takes
on no initial borrowing is lower than
µ+fi
fl+‰+µ+fi≠R
, she will find it not profitable to take on any
initial borrowing. If the marginal return from long-term investment when she takes on her initial
borrowing up to the debt limit is higher than
µ fl+‰+µ+fi
R fl+µ+‰ yh
limit.
≠
µ+fi
R ,
she will borrow up to her debt
Given the initial borrowing, the duration between an agent’s initial borrowing and reaching the
debt limit, T (a, b), depends on her initial borrowing and asset holding. It is solved by equation
a
h
= beRT (a,b) . So, T (a, b) =
1
R
ln
back the debt is
ˆ
T (a,b)
0
(µ + fi)e≠(‰+µ+fi)t dt =
!a"
hb . Given T (a, b), the probability that an agent is able to pay
S
µ+fi
µ+fi U
1 ≠ e≠(‰+µ+fi)T (a,b) =
1≠
‰+µ+fi
‰+µ+fi
Therefore, the probability of default is
Ë
‰
‰+µ+fi
È
+
µ+fi
‰+µ+fi
between initial borrowing and the debt limit, b̂ =
hb
a .
1
hb
a
2 ‰+µ+fi
R
3
hb
a
4 ‰+µ+fi
R
T
V.
, which is increasing in the ratio
From equation (14), b̂ is increasing in the
marginal return from project investment, f Õ (s + m + b), and decreasing in haircut and dividends
from an asset. This reflects the tradeoff between the return from project investment and losing
assets through default. So the default probability of an agent is increasing in the productivity of
the long-term technology and decreasing in haircut and the dividends of an asset.
The default probability of an agent after initial borrowing also depends on the liquidity of an
agent’s portfolio. For an agent with portfolio (a + hsÕ , m, s ≠ sÕ ), the following corollary shows that
the default probability is increasing in sÕ .
Corollary 6.1. Default triggers more default. Counter-party default increases default probability
of lenders in the future. For an agent with portfolio (a + hsÕ , m, s ≠ sÕ ),
db̂
>0
dsÕ
Corollary 6.1 is derived from Lemma 6.4. The portfolio of an agent with portfolio (a, m, s)
turns into (a + hs, m, 0) after she meets a defaulting agent. So, according to the corollary, an agent
whose counterparty defaults before she starts her long-term investment is more likely to default
than an agent who has not met a default counterparty when she starts her long-term investment.
31
Figure 9: intertemporal chain of borrowing and lending
This implies that counterparty default has an externality on the other agent’s default probability.
Figure 9 illustrates this externality. Agent i borrows from agent j and delivers her asset holding
to agent j upon default. This increases agent j’s default probability when she borrows from agent
k, among other agents, to invest in her long-term project. So, agent j is more likely to deliver
her assets to agent k upon default. As we can see from the inter-temporal chain of reactions, the
default of one agent increases the default probability of those agents who lend to her, which may
affect the default probability of those who lend to agent j’s. This intertemporal chain is reminiscent
of Kiyotaki and Moore [1997b], which studies the propagation of shocks through credit chains. In
my model, shocks to agents’ portfolio – making their portfolio less liquid – are contagious. The
increase in the probability of default passes on from a defaulting borrower to her lenders or, so to
speak, borrowers-to-be. The risk-taking of an individual borrower, therefore, adds to the risk of
the whole system. I will explore this type of contagion more in the section on dynamics.
Proposition 6.1. A stationary equilibrium with debt rollover must satisfy conditions in Lemma
6.1 and Lemma 6.2. The equilibrium can be summarized by interest rate R, haircut h, default rate
32
”, the portfolio choice of active lenders, (m, s), the initial borrowing of active lenders, b1 and the
initial borrowing of deactivated lenders, b0 , in the following system of equations,
3
3
4(fl+‰+µ+fi≠R)/R
hb1
a0
hb0
a0 + hs
4(fl+‰+µ+fi≠R)/R
µ+fi
f Õ (m0 + b1 ) ≠ fl+‰+µ+fi≠R
R
=
,
µ
µ+fi
fl + ‰ + µ + fi fl+µ+‰
yh ≠ fl+‰+µ+fi≠R
(15)
=
(16)
µ+fi
f Õ (m + b0 ) ≠ fl+‰+µ+fi≠R
R
,
µ
µ+fi
fl + ‰ + µ + fi fl+µ+‰
yh ≠ fl+‰+µ+fi≠R
µ (yh ≠ 1) s = sR + ⁄ [U (a, m, s) ≠ U (a + hs, m, 0)] ,
”=
Bd
,
B
(18)
1
s = B,
µ+⁄+”
(m, s) œ arg
(17)
(19)
max
m+sÆm0 ,mØ0,sØ0
V (a, m, s).
(20)
where B denotes total borrowing, B d denotes the borrowing from defaulting borrowers.
⁄
B=
b1
µ+⁄+”
ˆ
T1
0
(R≠µ≠fi)s
e
”
⁄
ds +
b0
µ+⁄µ+⁄+”
ˆ
0
T0
e(R≠µ≠fi)s ds,
and
Bd =
where T0 =
1
R
ln
1
hb0
a0 +hs
2
⁄
”
⁄
b1 e(R≠µ≠fi)T1 +
b0 e(R≠µ≠fi)T0 ,
µ+⁄+”
µ+⁄µ+⁄+”
and T1 =
1
R
ln
1
hb1
a0
2
.
Equations (15) and (16) are the optimality conditions for the initial borrowing of active and
deactivated lenders when they find their long-term projects. Equation (17) is the condition for
lenders to be willing to roll over the debt. Equation (18) pins down the equilibrium default rate,
which is the ratio of the demand for funding from defaulting borrowers and the total demand of
borrowers. Equation (19) is the market-clearing condition. Equation (20) pins down the portfolio
choice of active lenders.
6.2
The effect of collateral risk
As the equilibrium of the full model depends on two key features – collateral risk and maturity
mismatch – let us take one step back before looking at the comparative statics of the full model.
In this subsection, I study the effect of collateral risk on liquidity hoarding and cash in the repo
market by studying a state-contingent debt contract with an exogenous borrowing constraint. By
33
shutting down maturity mismatch, we can focus on the effect of collateral risk on the equilibrium
outcome. In the next subsection, I will go one step further to characterize the whole equilibrium
with maturity mismatch because of using short-term repo contracts.
6.2.1
Equilibrium with state-contingent repo contracts
The repayment of the repo contract I consider in this subsection is contingent on the arrival of
liquidity when the borrower’s collateral asset or long-term project matures.24 With state-contingent
repo contracts, there are two symmetric equilibria, one in which no agents will default when they
start borrowing and another in which all agents default immediately when they start borrowing.
Ó
Proposition 6.2. Suppose that min f Õ (m0 +
µy
a0
h ), fl+‰+µ h
Ô
>
(µ+fi)(1+R)
fl+‰+µ+fi ,
then, there exists a
unique stationary equilibrium with the state-contingent repo contract characterized by the following
system of equations,
‰
⁄
f Õ (m),
µ+fifl+⁄
⁄
a0
m0 ≠ m =
,
µ + fi (1 + R)h
R=
(21)
(22)
Proof of the proposition is left to the Appendix. The assumption in the proposition ensures
that an agent who borrows to invest in her long-term project would borrow against all her collateral
assets because the return from project investment is high enough. According to proposition 6.2, the
interest rate increases in equilibrium liquidity hoarding of agents waiting for their own investment
opportunities, taking parameter values as given. In the next proposition, I do some comparative
statics.
Proposition 6.3. Under the assumption in Proposition 6.2,
1.
ˆm
ˆ‰
> 0,
ˆR
ˆ‰
> 0,
ˆi
ˆ‰
> 0;
2.
ˆm
ˆfi
> 0,
ˆR
ˆfi
< 0,
ˆi
ˆfi
> 0.
Proposition 6.3 gives the comparative statics with respect to collateral risk and the maturity
of the project. When collateral risk increases, agents waiting for their investment opportunity
24
Without loss of generality, I assume that the liquidity at the maturity of either the project or the collateral asset
is enough to repay all the debt.
34
refrain from inter-bank lending and increase storage, as consumptions good in storage would be
the only resource available when the economy is in the bad state. As a result, funding for the
repo market dries up. Thus, the interest rate of the repo contract increases and investment in
long-term investment drops. When the maturity of the project decreases, liquidity from maturing
projects flows more frequently to lenders and, subsequently , results in more project investment and
liquidity hoarding. With more funding available to lenders, the tension between liquidity hoarding
and project investment is tempered. Consequently, both liquidity hoarding and investment increase
and interest rates drop.
6.3
Market liquidity, solvency and balance sheet of financial intermediaries
In this subsection, I examine the comparative statics of the rollover equilibrium, keeping the expected payoff from dividends from collateral fixed.25 This way, the effect of collateral risk on the
rollover equilibrium will be clearer.26
Figure 10 illustrates how collateral risk affects the liquidity of the repo market and the solvency
of financial intermediaries. Figure 11 illustrates how collateral risk affects the aggregated balance
sheet of financial intermediaries. When the arrival rate of devaluation shock increases, banks hoard
more liquidity in their storage account and refrain from lending to other banks. This increases the
interest rate and haircut in the market, which is consistent with the analytical results we derive
in an environment without maturity mismatch. With maturity mismatch between investment and
liability, borrowers are more likely to default, facing the higher haircut and interest rates. The
increase in default rate would further increase the haircut and interest rates.
The disfunctioning of the repo market is reflected in the aggregate long-term investment and
leverage ratio, as collateral risk, the long-term investment and therefore the aggregate output of the
economy drops. And Figure 11 shows that repo lending contributes less to long-term investment,
and more investment uses funding from storage.
In Figure 11, the increase in storage, which represents liquidity hoarding, and the decrease in
repo lending as collateral risk increases are not completely substitutable. As collateral risk increases,
default is more likely, rollover is harder and lenders have more illiquid collateral asset but less liquid
25
When I increase devaluation shock ‰, I also increase dividend payment from a collateral tree when the economy
y
y0
is in the good state. So the dividend payment y satisfies the following condition, fl+µ+‰
= fl+µ+‰
.
0
26
Parameter values used in the comparative statics and transition dynamics is listed in Table 2 in the appendix.
35
haircut (percentage)
interest rate
0.12
0.1
0.08
0.06
0.04
0.02
0
40
30
20
10
0
0.01
0.02
0.03
0.04
0.01
0.02
0.03
0.04
default rate
0.1
0.08
0.06
0.04
0.02
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
arrival rate of devaluation shock
Figure 10: Collateral risk and the liquidity of the repo market and solvency of financial intermediaries.
asset. As a result, the sum of repo lending and storage, which is what we call short-term asset,
decreases. Notice that with the state-contingent repo contract, the sum of repo lending and reserve
is constant. The decrease in reflects the increase in default rate. When debt rollover is harder, the
value of repo lending drops and illiquid collateral asset held by agents waiting for their investment
opportunities increases.
While comparative statics in the model resemble a Kiyotaki-Moore type of model assuming
exogenous borrowing constraint, the constraint in my model is endogenous and depends on the
collateral risk and the contract structure. For example, the haircut for the overnight repo contract
would be higher than that for the state-contingent repo contract. The endogenous haircut is
especially relevant when we study equilibrium dynamics, where the borrowing constraint implied
from haircut will move endogenously along the dynamic path. I now turn to characterize the
transition dynamics triggered by an unexpected shock to the riskiness of collateral assets.
36
loan to collateral
LT investment
3.5
3.4
3.3
3.2
3.1
3
0.24
0.22
0.2
0.18
0.16
0.14
0.01
0.02
0.03
0.04
0.01
0.02
0.03
0.04
repo lending
hoarding
short−term asset
5
4
3
2
1
0
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
Figure 11: Credit risk and the aggregated balance sheet of the securitized banking system.
7
Equilibrium dynamics
In this section, I characterize the transition dynamics when agents’ expectations about the collateral
risk become more pessimistic. The exercise is intended to capture the transition dynamics triggered
by a shock similar to the one that we experienced during the great recession. More importantly, it
will help us understand the stability of the shadow banking system to shocks to collateral risk.
I assume that the economy before t = 0 is the steady-state equilibrium with Êt = 1, with the
parameter values given in Table 2. At t = 0, ‰ switches from 0.01 to 0.02. The algorithm to
compute the transition dynamics is in the Appendix.
Figure 13 shows the transition dynamics for the liquidity of the repo market and the solvency
of financial intermediaries. Figure 14 illustrates the transition dynamics for the aggregated balance
sheet of financial intermediaries in the securitized banking system. And Figure 15 illustrates the
transition dynamics of output and investment and the impact of the financial crisis on the real
economy.27
27
In Figure 14 and Figure 15, I normalized the values of the variables in the initial steady state to one.
37
The stability can be measured by two dimensions: initial response and persistence of the impact.
Initial response of the shadow banking system
Let’s first look at the initial response of the economy to the shift in expectations. At the moment
the expectations shift, haircut increases discontinuously and overshoots to a level even higher than
the haircut in the new steady state. This means that the debt limit that borrowers face given their
collateral asset holding drops discontinuously. As a result, borrowers with debt holdings above the
new debt limit but below the old debt limit are forced to default at t. The mass of default drains
the funding available in the repo market. Thus, the aggregate amount of repo lending drops, and
the massive initial default shows up in the discontinuous drop in repo lending in Figure 14. The
overshooting of haircut increases the probability of the initial default. The figure shows that about
25% of outstanding repo borrowing defaults at the moment the shock hits the economy.
The hike in haircut reflects the fact that the massive default drains liquidity from the market
as lenders’ liquid funding is replaced by illiquid collateral assets from defaulting borrowers. With
less funding available for both liquidity hoarding and repo lending, the interest rate overshoots.
As new entrants bring in more liquidity over time, the interest rates and haircut drops, and
liquidity hoarding increases. But the crisis is not over yet. The counterparty default risk remains
high. The overshooting in the default rate drains the financial system’s liquidity and triggers more
default in the future. This leads us to the persistence of the crisis.
Persistence of the impact of the shock
Massive initial default and high default rate related to the debt overhang for borrowers who started
borrowing before the crisis leads to more default in the future. We can see that the overshooting in
drop in investment, repo borrowing or lending and depressed liquidity hoarding persist for a long
period. If we think of a period as a quarter, it would still take more than ten years for the economy
to recover from the crisis and recover to the output level in the new steady state.
The disruption in the financial system has a big impact on investment and output of the
economy. Figure 15 shows that total investment drops by 20% immediately after the shock to
collateral risk and eventually drops by about 12%. The aggregate output responds with delay
because it takes time for investment in new projects to yield output. Because of the contagion
38
effect and the initial massive default, the decrease in output also overshoots. The maximum drop
is about 7%, after which the output recovers a little, and the output at the new steady state is
about 5% lower than at the old steady state.
The fluctuation on the transition path is related to the massive initial default and the outstanding repo borrowing when the shock hits the economy. Before the shock, investors borrow
more without worrying too much about default because of lower interest rates and lower haircut.
When collateral risk increases unexpectedly, borrowers with outstanding repo borrowing appear
to be over-borrowing and, therefore, likely to default under the higher haircut and interest rate.
The hike in the default rate leads to fluctuations in market liquidity and the balance sheet of the
shadow banking system. Figure 12 illustrates the contagion of counterparty default. It shows two
components of the demand for repo borrowing: one (the blue line) from investors who are still
waiting for their investment opportunities but have illiquid portfolios (more collateral asset and
less consumption goods than their initial endowment) at time zero, another (the red line) from
the rest of the borrowers, including those who started borrowing before time zero and were not
forced to default at time zero, those with liquid portfolio before they start borrowing and those
with illiquid portfolio before start borrowing because they experienced counterparty default after
time zero. The first component first increases and then, declines, with a sharp turn. The turn
takes place when investors whose demand for repo borrowing is included in the first component
start to default. The peak helps us see more clearly the effect of intertemporal contagion through
default. Since default triggers more default, default rate for this group of investors remain high.
And because of the massive initial default, the high default rate of this group shows up as a spike
in the aggregate default rate shown in Figure 13. The second component includes those investors
who started borrowing before time zero and were not forced to default at time zero. The demand
from those investor constitutes a debt overhang problem: Since they started borrowing when the
economy was in the initial steady-state equilibrium, in which agents expect that collateral risk is
low, they borrow more than they would have if they had realized the risk; this means the default
rate for this group of borrowers with over-hang debt will be high. Since there are two types of
borrowers with different initial borrowings and debt limits in the steady state before the shock
to expectation hits the economy, the distribution of borrowers is a combination of two continuous
distributions. This is why we can see two distinct peaks in default rate at the beginning of the
39
4
3.5
3
demand
2.5
2
1.5
1
0.5
0
−0.5
0
10
20
30
40
50
60
70
80
time
Figure 12: Decomposition of repo demand
transition path, before investors in the first group start to default.
Overall, the debt overhang and initial massive default lead to fluctuations in default rate and the
hikes in default rate after time zero leads to more default on the dynamic path. The contagion of
counterparty default and illiquidity of lenders’ portfolio leads to persistent overshooting in default
rate, haircut, interest rates, leverage ratio, investment and output.
After the breakout of crisis in 2007, the Federal Reserve stepped in to provide liquidity to the
dealer banksAdrian et al. [2009]. In daily news, many argue whether bailing out “too big to fail”
dealer banks is a good idea. From this exercise, we can see a counterfactual equilibrium dynamics
for what would have happened after the shift in expectation when the Federal Reserve Bank had
not stepped in. The mass of default would trigger over-shooting in haircut and interest rate and
misallocation and fluctuation of lenders’ funding between repo lending and reserve.
40
haircut h (%)
30
25
20
0
10
20
30
40
50
30
40
50
30
40
50
interest rate R
time
0.12
0.1
0.08
0.06
0
10
20
time
default rate
0.14
0.12
0.1
0.08
0.06
0
10
20
time
Figure 13: Equilibrium dynamics of the liquidity of the repo market and solvency of financial
intermediaries, when collateral risk increases unexpectedly.
41
loan to collateral
0.22
0.21
0.2
0.19
0.18
0.17
0.16
0
10
20
30
40
50
30
40
50
liquidity hoarding
time
1.8
1.6
1.4
1.2
1
0
10
20
time
Figure 14: Equilibrium dynamics of the aggregate balance sheet of the securitized banking system,
when collateral risk increases unexpectedly.
42
investment
1
0.95
0.9
0.85
0.8
0
10
20
30
40
50
30
40
50
time
output
1
0.98
0.96
0.94
0.92
0
10
20
time
Figure 15: Equilibrium dynamics of investment and output, when collateral risk increases unexpectedly.
43
8
Conclusion
In this paper, I build a dynamic model to study the efficiency and stability of the shadow banking
system. I show that collateral risk leads to an increase in counterparty default risk in the equilibrium
with repo rollover. Counterparty default drains liquidity from the repo market and reduces output.
By studying the dynamic equilibrium triggered by a shock to collateral risk, I show that the shift
in collateral risk could be an important contributor to the disruptions in the repo market and the
shadow banking system we observed during the 2007-2008 financial crisis. And the shadow banking
system is vulnerable to shifts in market participants’ perception of the collateral risk in two senses.
First, a small shift in the market belief could trigger a massive initial default. Second, the effect
of the shock is long-lasting. The exercise shows that the externality of counterparty default has
important implications for the efficiency and stability of the shadow banking system and related
government policies.
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crisis. IMF Economic Review, 58(1):118–156, 2010.
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2012.
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Quarterly Review, pages 37–53, 2008.
Timothy J Kehoe and David K Levine. Debt-constrained asset markets. The Review of Economic
Studies, 60(4):865–888, 1993.
Nobuhiro Kiyotaki and John Moore. Credit cycles. Journal of Political Economy, 105(2), 1997a.
Nobuhiro Kiyotaki and John Moore. Credit chains. 1997b.
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92(2):62–66, 2002.
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1(1), 2001.
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Economic Research, 2012.
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Journal of Political Economy, 113(3):463–484, 2005.
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Paper, 2011.
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48
parameter
value
parameter
value
fl
0.03
µ
0.033
⁄
0.09
fi
0.1
‰
0.01
y
1
◊
4
–
0.38
a0
4.17
m0
1
Table 2: Parameter values used in numerical exercises
A
Laws of motion
First if Êt = 1, the law of motion of F0t (a, m, s) is
1
dF0t
(a, m, s)
Q
aÕ Æ a,
R
c
d
c
d
1
I c m̃0t (aÕ , mÕ , sÕ ) Æ m, d (µ + ⁄) dt(1 ≠ d˜0t (aÕ , mÕ , sÕ ))dF0,t≠dt
(aÕ , mÕ , sÕ )
a
b
s̃0t (aÕ , mÕ , sÕ ) Æ s
Q
R
Õ
a
Æ
a,
ˆ c
d
c
d
1
≠ I c mÕ Æ m, d d˜0t (aÕ , mÕ , sÕ )dF0,t≠dt
(aÕ , mÕ , sÕ )
a
b
sÕ Æ s
Q
R
Õ
a
Æ
a,
ˆ c
d
c
d
1
+ I c m̃0t (aÕ , mÕ , sÕ ) Æ m, d ”t IsÕ >0 dt (1 ≠ z̃0t (aÕ , mÕ , sÕ )) dF0,t≠dt
(aÕ , mÕ , sÕ )
a
b
s̃0t (aÕ , mÕ , sÕ ) Æ s
Q
R
Õ
Õ
a
+
h
s
Æ
a,
t
ˆ c
d
c
d
1
+ I c m̃0t (aÕ , mÕ , sÕ ) Æ m, d d˜0t (aÕ , mÕ , sÕ )dF0,t≠dt
(aÕ , mÕ , sÕ )
a
b
s̃0t (aÕ , mÕ , sÕ ) Æ s
Q
R
a Æ a0 ,
c
d
c
d
+ ÷dtI c m̃0t (a0 , 0, m0 ) Æ m, d
a
b
s̃0t (a0 , 0, m0 ) Æ s
=≠
ˆ
(23)
The first component in equation 23 is the outflow of maturing assets and agents receiving an
investment opportunity, conditional on the agent not defaulting. Agents with maturing assets
consumes and leave the economy. Agents with incoming projects flow to the measure F1t . The
second component is the outflow of defaulting agents from type (aÕ , mÕ , sÕ ) to other types. The third
component is the inflow of agents who meet defaulting borrowers at t ≠ dt but do not accept them.
49
The fourth component is the inflow from defaulting agents. The last component is the inflow from
newcomers at t ≠ dt.
Q
aÕ Æ a,
R
c
d
c
d
1
I c m̃dt (aÕ , mÕ ) Æ m, d (µ + ⁄) dtdF0,t≠dt
(aÕ , mÕ )
a
b
s̃dt (aÕ , mÕ ) Æ s
Q
R
Õ
Õ
Õ Õ
a
+
(1
+
h)s̃
(a
,
m
,
s
)
Æ
a,
0t
ˆ c
d
c
d
Õ
Õ
Õ
+ I c m̃0t (a , m , s ) Æ m,
d ”t IsÕ >0 dtz̃0t (aÕ , mÕ , sÕ )dF0,t≠dt (aÕ , mÕ , sÕ )
a
b
s̃0t (aÕ , mÕ , sÕ ) Æ s
1
dFdt
(a, m) = ≠
ˆ
Similarly, if Êt = 1, the law of motion of F1t (a, m, s, i) is
50
(24)
dF1t (a, m, s, i) = ≠
ˆ
Q
aÕ Æ a,
c
c
c m̃1t (aÕ , mÕ , sÕ ) Æ m,
Ic
c
c s̃1t (aÕ , mÕ , sÕ ) Æ s,
a
iÕ Æ i
Q
R
d
d
d
d (µ + fi) dt(1 ≠ d˜1t (aÕ , mÕ , sÕ , iÕ ))dF1,t≠dt (aÕ , mÕ , sÕ , iÕ )
d
d
b
(25)
R
aÕ Æ a,
c
d
d
ˆ c Õ
c m Æ m, d
d d˜1t (aÕ , mÕ , sÕ , iÕ )dF1,t≠dt (aÕ , mÕ , sÕ , iÕ )
≠ Ic
c Õ
d
c s Æ s, d
a
b
iÕ Æ i
Q
R
aÕ + ht s̃1t (aÕ , mÕ , sÕ , iÕ ) Æ a,
c
d
d
ˆ c
c m̃1t (aÕ , mÕ , sÕ , iÕ ) Æ m,
d
d ”t IsÕ >0 dtz̃1t (aÕ , mÕ , sÕ , iÕ )dF1,t≠dt (aÕ , mÕ , sÕ , iÕ )
+ Ic
c
d
Õ
Õ
Õ
Õ
c s̃1t (a , m , s , i ) Æ s,
d
a
b
Õ
i Æi
Q
R
aÕ Æ a,
c
d
d
ˆ c
c m̃1t (aÕ , mÕ , sÕ ) Æ m, d
d ”t IsÕ >0 dt (1 ≠ z̃1t (aÕ , mÕ , sÕ , iÕ )) dF1,t≠dt (aÕ , mÕ , sÕ , iÕ )
+ Ic
c
d
Õ
Õ
Õ
c s̃1t (a , m , s ) Æ s, d
a
b
iÕ Æ i
Q
R
aÕ + (1 + h)sÕ Æ a,
c
d
d
ˆ c
c m̃1t (aÕ , mÕ , sÕ , iÕ ) Æ m, d
d d˜1t (aÕ , mÕ , sÕ , iÕ )dF1,t≠dt (aÕ , mÕ , sÕ , iÕ )
+ Ic
c
d
Õ
Õ
Õ
Õ
c s̃1t (a , m , s , i ) Æ s, d
a
b
iÕ Æ i
Q
R
aÕ Æ a,
c
d
d
ˆ c
c ĩIt (aÕ , mÕ , sÕ ) Æ i,
d
d dF0,t≠dt (aÕ , mÕ , sÕ )
+ ⁄dt I c
c
d
Õ
Õ
Õ
c m̃It (a , m , s ) Æ m, d
a
b
s̃It (aÕ , mÕ , sÕ ) Æ s
Q
R
aÕ Æ a,
c
d
d
ˆ c
c ĩIt (aÕ , mÕ , 0) Æ i,
d
c
d dFd,t≠dt (aÕ , mÕ )
+ ⁄dt I c
(26)
d
c m̃It (aÕ , mÕ , 0) Æ m, d
a
b
s̃It (aÕ , mÕ , 0) Æ s
1 (Œ, Œ, Œ) = 0, F 1 (Œ, Œ) = 0,F 1 (Œ, Œ, Œ, Œ) = 0 , and
If Êt≠dt = 0, F0t
1t
dt
51
dG0t (m) = ≠ (µ + ⁄) dt
dG1t (m, i) = ≠fidt
ˆ
{m̃I0 (mÕ )Æm}
ˆ
{m̃(mÕ )Æm}
dG0t≠dt (mÕ ) + ÷dtI;
dG1,t≠dt (m , i) + ⁄dt
Õ
ˆ
ˆ
I;
I;
mÕ Æ m
+ ÷dtI;
G1t (m, i) =
ˆ
+ ⁄dt
Efficiency
] ĩI0 (mÕ ) Æ i,
< [1 ≠ (µ + ⁄) dt] dF
< [1 ≠ (µ + ⁄) dt] dF
m̃0A (w) Æ m
0,t≠dt
Õ
Z dG
0t≠dt (m )
_
^
d,t≠dt
! Õ Õ Õ"
a ,m ,s
ˆ
IY
_
] m̃0A
(mÕ )
(29)
<
! Õ Õ Õ Õ"
Z [1 ≠ (µ + fi) dt] dF
a ,m ,s ,i
1,t≠dt
] mÕ Æ m, _
^
_ Õ
_
[
\
i Æi
ˆ
(28)
! Õ Õ"
a ,m
IY
_
+ ⁄dt
B
mÕ Æ m
IY
_
(27)
_
[ m̃I0 (mÕ ) Æ m _
\
If Êt≠dt = 1, then conditional on the Êt = 0,
G0t (m) =
ˆ
m̃0A Æ m
<
Õ
Õ Õ
Z dF
0,t≠dt (a , m , s )
_
Æ m, ^
_
[ ĩ0A (mÕ ) Æ i
_
\
_
[ ĩ0A (mÕ ) Æ i
_
\
Õ
Õ
Z dF
d,t≠dt (a , m )
] m̃0A (mÕ ) Æ m, _
^
IY
_
Proof for Proposition 5.1:
52
(30)
Proof.
St =
max
{C· ,m· ,i· }’· Øt
C· d· + (⁄d· )
ˆ
Œ
C· e≠fl(· ≠t) d·,
t
÷
µ+⁄ i·
+ m·
s.t.
C· , m· , i·
fl+fi –
≠fi(· ≠s) ⁄ ÷ ds
µ+⁄
≠Œ fi ◊is (fid· ) e
´·
+ ≠Œ ayÊ· (µd· ) e≠µ(· ≠s) ÷ds + (÷d· ) M·
Æ
´·
Ø 0.
e≠fl(· ≠t) “·
+ m· ≠d· ,
“c· , “l· , “m· , “i·
FOC
C· :
e≠fl(· ≠t) d· ≠ e≠fl(· ≠t) “· d· + “c·
=0
÷
◊–i–≠1
(fl + fi) d· e≠fi(s≠· ) ⁄ µ+⁄
e≠fl(s≠t) “s ds + “i·
·
=0
≠e≠fl(· ≠t) “· + e≠fl(· +d· ≠t) “· +d· + “m·
m· :
i· :
÷
≠⁄d· µ+⁄
e≠fl(· ≠t) “· +
´Œ
·
≠ 1 + ◊–i–≠1
·
◊–i–≠1
·
ˆ
Œ
=0
(fl + fi) e≠(fl+fi)(s≠· ) ds = 0
·
=1
Substituting the resource constraint to the objective function ...
5
6
fl + fi – ≠fi(· ≠s)
÷
◊is fie
⁄
ds +
fi
µ+⁄
≠Œ
fl+fi –
÷
=
◊is ⁄
+ a÷
fi
µ+⁄
5
6
fl + fi ú–
⁄÷
C ú ≠ ÷mú =
◊i ≠ iú
+ a÷
fi
µ+⁄
÷
Ct ≠ ÷mt + ⁄
it =
µ+⁄
C
C.1
ˆ
·
ˆ
·
aµe≠µ(· ≠s) ÷ds
≠Œ
the effect of collateral risk
Equilibrium with state-contingent collateralized debt contract and exogenous haircut
The contract is contingent on liquidity arrival: At the date of borrowing, the borrower puts down h
units of collateral for each unit of consumption good she borrows; when the borrower has liquidity
53
to repay the debt, the repayment is 1 + R, and if the borrower does not repay the debt, the lender
will keep the collateral.
The borrower’s expected payoff when she invests in a long-term project is
U (a0 , m, s) = max f (m + s + b) +
b
s.t.0 Æ b Æ
µy
(µ + fi)(1 + R)b
a0 ≠
fl+‰+µ
fl+‰+µ+fi
a0
(1 + R)h
FOC:
f Õ (m + s + b) ≠ “1 ≠ “0 ≠
(µ + fi) (1 + R)
=0
fl+‰+µ+fi
If the optimal borrowing is such that f Õ (m + s + b) >
(µ+fi)(1+R)
fl+‰+µ+fi ,
a0
(1+R)h .
Otherwise,
Assume that ◊ is high enough and a0 and m0 are small enough so that, f Õ (m+s+b) >
(µ+fi)(1+R)
fl+‰+µ+fi ,
f Õ (m + s + b) =
or f Õ (m + s +
then b =
(µ+fi)(1+R)
fl+‰+µ+fi .
a0
h)
>
(µ+fi)(1+R)
fl+‰+µ+fi ,
then,
3
U (a0 , m, s) = f m + s +
a0
h
4
+
5
6
µy
µ+fi
≠
a0
fl + ‰ + µ (fl + ‰ + µ + fi) h
The value function of an agent before she finds her long-term project is
⁄
f (m) + (µ + fi)Rs
fl+⁄
; 3
4 5
6 <
a0
µy
µ+fi
+⁄ f m+s+
+
≠
a0
h
fl + ‰ + µ (fl + ‰ + µ + fi) h
(fl + µ + ‰ + ⁄)V (a0 , m, s) = µ(a0 y + m0 ) + ‰
⁄
f (m) + (µ + fi)R (m0 ≠ m)
fl+⁄
; 3
4 5
6 <
a0
µy
(µ + fi)(1 + R)
+ ⁄ f m0 +
+
≠
a0
h
fl + ‰ + µ (fl + ‰ + µ + fi) h
(fl + µ + ‰ + ⁄)V (a0 , m, m0 ≠ m) = µ(a0 y + m0 ) + ‰
The optimal choice of portfolio implies that
R=
dV (a0 ,m,m0 ≠m)
dm
‰
⁄
f Õ (m)
µ+fifl+⁄
54
= 0.
The equilibrium is solved by the following system of equations:
‰
⁄
f Õ (m),
µ+fifl+⁄
m0 ≠ m
⁄
a0
1
=
,
µ+⁄
µ + ⁄ (1 + R)h µ + fi
R=
the last equation being the market-clearing condition.
With the system of equations, the net supply to the market for the state-contingent contract,
, can be reduced to a function of storage holding m.
⁄
a0
µ + fi (1 + R)h
⁄ a0
1
= m0 ≠ m ≠
‰
⁄
µ + fi h 1 + µ+fi fl+⁄
f Õ (m)
(µ + ⁄) (m) = m0 ≠ m ≠
⁄ a0
(0) = 0 and (µ + ⁄) (m0 ) = ≠ µ+⁄
h 1+
(µ + ⁄) Õ (m) = ≠1 +
1
‰
⁄
f Õ (m0 )
µ+fi fl+⁄
< 0.
‰
⁄
ÕÕ
⁄ a0
µ+fi fl+⁄ f (m)
Ó
Ô <0
µ + fi h 1 + ‰ ⁄ f Õ (m) 2
µ+fi fl+⁄
So, if the equilibrium exists, it is unique.
Comparative statics
(µ + ⁄)
1
⁄
Õ
ˆ
⁄ a0
µ+fi fl+⁄ f (m)
=
Ë
È >0
ˆ‰
µ + fi h 1 + ‰ ⁄ f Õ (m) 2
µ+fi fl+⁄
ˆ
⁄ a0
(µ + ⁄)
=≠
ˆfi
µ+fi h 1+
=
Y
]
Z
‰
⁄
Õ
≠ (µ+fi)
2 fl+⁄ f (m) ^
1
≠
≠
‰
‰
⁄
⁄
Õ
Õ
[ µ+fi
\
1 + µ+fi
µ+fi fl+⁄ f (m)
fl+⁄ f (m)
⁄
a0
2 h
1+
(µ + fi)
1
1
‰
⁄
Õ
µ+fi fl+⁄ f (m) 1
>0
55
1
+
‰
⁄
Õ
µ+fi fl+⁄ f (m)
D
Characterization of equilibrium with debt rollover
D.1
The problem of agents with a long-term project
Given the price of assets at fire sale, implied by h, the maximum amount of borrowing is
a0
h.
At the
µ
moment of default, the borrower’s continuation value is f (i) + fl+µ+‰
(a ≠ a0 ). For a borrower with
a assets decides to default when she puts down a0 units of asset as collateral, the value function
can be solved by the following differential equation. It is easy to verify that an agent will hold zero
storage after her long-term investment. So, we denote W (a, b, 0, i) = max0Æa0 Æa W̃ (a, b, a0 , i)
flW̃ (a, b, a0 , i) = fi
5
6
fl+fi
µ
f (i) +
ya ≠ b ≠ W̃ (a, b, a0 , i)
fi
fl+µ+‰
+ ‰ [f (i) ≠ W (a, b, a0 , i)]
3
4
+ µ [f (i) + ya ≠ b ≠ W (a, b, a0 , i)] +
ˆ W̃ (a, b, a0 , i)
bR,
ˆb
a0
µy
W̃ a, , a0 , i = f (i) +
(a ≠ a0 ),
h
fl+µ+‰
1
2
a0
where W̃ a, 1+h
, a0 , i is the continuation when she defaults.
Lemma D.1.
5
6
3
µ + fi ≠ ‰·
1
µ
hb
≠
y a0
fl+‰+µ+fi≠Rh fl+µ+‰
a0
µ + fi ≠ ‰·
µ
≠
b + f (i) +
ya
fl+‰+µ+fi≠R
fl+µ+‰
W̃ (a, b, a0 , i) =
4(fl+‰+µ+fi)/R
Proof.
(fl + ‰ + µ + fi) W̃ (a, b, a0 , i) = (fl + ‰ + µ + fi) f (i) ≠ (µ + fi)b + µ
1
Guess W̃ (a, b, a0 , i) = C0 b(fl+‰+µ+fi)/R + C1 b + C2 f (i) +
fl+‰+µ+fi
C0 b(fl+‰+µ+fi≠R)/R
R
+ C1 .
5
3
fl+µ+‰+fi
ˆ W̃ (a, b, a0 , I)
ya +
bR
fl+µ+‰
ˆb
2
µ
fl+µ+‰ ya
+ C3 . Then
4
ˆ W̃ (a,b,a0 ,I)
ˆb
µ
ya + C3
fl+µ+‰
fl+µ+‰+fi
=(fl + ‰ + µ + fi)f (I) ≠ (µ + fi ≠ ‰· )b + µ
ya
fl+µ+‰
5
6
fl+‰+µ+fi
+ bR
C0 b(fl+‰+µ+fi≠R)/R + C1
R
(fl + ‰ + µ + fi) C0 b(fl+‰+µ+fi)/R + C1 b + C2 f (I) +
56
6
=
(fl + ‰ + µ + fi) C1 = RC1 ≠ (µ + fi ≠ ‰· )
(fl + ‰ + µ + fi) C2 = (fl + ‰ + µ + fi)
(fl + ‰ + µ + fi) C3 = 0
µ+fi≠‰·
C1 = ≠ fl+‰+µ+fi≠R
, C2 = 1, C3 = 0.
1
2
µ+fi≠‰·
µ
a0
Therefore, W̃ (a, b, a0 , I) = C0 b(fl+‰+µ+fi)/R ≠ fl+‰+µ+fi≠R
b+f (i)+ fl+µ+‰
ya. Since W̃ a, 1+h
, a0 , i =
f (i) +
C0
3
µy
fl+µ+‰ (a
a0
h
≠ a0 ), we have
4(fl+‰+µ+fi)/R
≠
µ+fi
a0
µ
µy
+ f (i) +
ya = f (i) +
(a ≠ a0 )
fl+‰+µ+fi≠R h
fl+µ+‰
fl+µ+‰
5
6
3
µ+fi
1
µ
a0
C0 =
≠
y a0
fl+‰+µ+fi≠Rh fl+µ+‰
h
5
6
3
µ+fi
1
µ
hb
W̃ (a, b, a0 , i) =
≠
y a0
fl+‰+µ+fi≠Rh fl+µ+‰
a0
µ
µ+fi
≠
b + f (i) +
ya
fl+‰+µ+fi≠R
fl+µ+‰
4≠(fl+‰+µ+fi)/R
4(fl+‰+µ+fi)/R
Lemma D.2. The partial derivative of the value function is
C
3
ˆ W̃ (a, b, a0 , i)
µ
hb
=
a ≠ a0
ˆy
fl+µ+‰
a0
5
4(fl+‰+µ+fi)/R D
6
ˆ W̃ (a, b, a0 , i)
µ
µ+fi
1 fl+‰+µ+fi≠R
=
y≠
ˆa0
fl+µ+‰
fl+‰+µ+fi≠Rh
R
ˆ W̃ (a, b, a0 , i)
µ
=
y
ˆa
fl+µ+‰
5
6
C
3
3
hb
a0
ˆ W̃ (a, b, a0 , i)
µ
µ+fi
fl+‰+µ+fi
hb
=
yh ≠
≠
ˆb
fl+µ+‰
fl+‰+µ+fi≠R
R
a0
µ+fi
≠
fl+‰+µ+fi≠R
ˆ W̃ (a, b, a0 , i)
= f Õ (i)
ˆi
4(fl+‰+µ+fi)/R
4(fl+‰+µ+fi≠R)/R D
From Lemma D.2, we can see that the borrower will roll over her debt up to the debt limit as
long as
µ
fl+µ+‰ y
≠
µ+fi
1
fl+‰+µ+fi≠R h
> 0. And, in this case, W (a, ≠b, i) = W̃ (a, b, a, i).
57
Lemma D.3. agents
• will borrow against all their asset if
• will not borrow if h <
µ
fl+µ+‰ y
≠
> 0 or h >
µ+fi
fl+µ+‰
fl+‰+µ+fi≠R µy .
µ+fi
fl+µ+‰
fl+‰+µ+fi≠R µy
• are indifferent between borrowing or not if h =
D.2
µ+fi
1
fl+‰+µ+fi≠R h
µ+fi
fl+µ+‰
fl+‰+µ+fi≠R µy .
The problem of agents when they find a long-term project
Next, I consider the optimization problem at the moment the agent finds an investment opportunity.
U (a, m, s) =
max
0
cØ0,0ÆbÆ 1+h
a
c + W (a, ≠b, s + m ≠ c + b)
The first-order condition of the problem is
dW (a, ≠b, s + m ≠ c + b)
= 0,
db
ˆ
ˆ
W + W = 0.
ˆb
ˆi
From the first-order condition we have the following lemma.
Lemma D.4. The optimal choice of project investment and initial borrowing of an agent with
portfolio (a, s, m) is solved by equation
3
Let b̂ =
hb
a+hsÕ .
hb
a
b=
4(fl+‰+µ+fi≠R)/R
a+hsÕ
h b̂,
=
.
3
Ë
µ+fi
f Õ (s + b) ≠ fl+‰+µ+fi≠R
R
µ
µ+fi
fl + ‰ + µ + fi fl+µ+‰
yh ≠ fl+‰+µ+fi≠R
s ≠ sÕ + b = s + ha b̂ ≠ (1 ≠ b̂)sÕ . Let
µ
fl+µ+‰ yh
Ë
=
1
≠
≠ f Õ s + ha b̂ ≠ (1 ≠
È
fl+‰+µ+fi (fl+‰+µ+fi≠R)/R
b̂
R
È
µ+fi
b̂)sÕ ≠ fl+‰+µ+fi≠R
µ+fi
fl+‰+µ+fi≠R
2
4
ˆ
a
= (1 ≠ b̂)f ÕÕ s + b̂ ≠ (1 ≠ b̂)sÕ < 0
Õ
ˆs
h
5
6
ˆ
µ
µ+fi
fl + ‰ + µ + fi fl + ‰ + µ + fi ≠ R (fl+‰+µ+fi≠2R)/R
=
yh ≠
b̂
‰
fl+‰+µ+fi≠R
R
R
ˆ b̂ 3 fl + µ +
4
3
4
a
a
≠
+ sÕ f ÕÕ s + b̂ ≠ (1 ≠ b̂)sÕ > 0
h
h
Then, from the Implicit Function Theorem, we have the following result:
58
Lemma D.5. Counterparty default increases the default probability of lenders in the future
db̂
>0
dsÕ
Since default probability is an increasing function of b̂, we know from this lemma that counterparty default that transforms liquid funding to collateral assets will increase the default probability
when the agent starts borrowing.
D.3
The problem of agents waiting for an investment opportunity
In the stationary environment, the value functions of agents waiting for their investment opportunities, given the optimal portfolio choice on storage m and lending s, can be written as follows:
(fl + µ + ‰ + ⁄ + ”) V (a, m, s) = sR + µ (ya + s + m) + ”V d (a + hs, m) + ‰V A (m) + ⁄U (a, m, s),
(fl + µ + ‰ + ⁄) V d (a + hs, m) = µ [y (a + hs) + m] + ‰V A (m) + ⁄U (a + hs, m, 0).
(fl + ⁄)V A (m) = ⁄◊m–
Lemma D.6. The following condition must be satisfied in equilibrium:
µ (yh ≠ 1) s =sR + ⁄ [U (a, s, m) ≠ U (a + hs, 0, m)]
Proof. Suppose, instead, that V d (a + hs, m) > V (a, m, s). Then lenders will default on the loan
by keeping the collateral. So, V d (a + hs, m) Æ V d (a, m, s). Suppose, V d (a + hs, m) < V (a, m, s).
Then, lenders won’t be willing to lend to the defaulting borrowers when they observe that the
borrowers are going to default. Rollover is not possible.
Given the value functions, the optimal portfolio choice is solved by the following problem:
max V (a, m, s)
s,mœR+
s.t.s + m Æ m0
59
E
Equilibrium dynamics with a constant ‰
Given the initial condition, the distribution of lenders and borrowers, we need to solve the sequence
of default rate”t , interest rate Rt , haircut ht , and the optimal portfolio choice of active lenders
between repo lending st and storage mt .
Active lenders’ portfolio choice
Given the value functions,
sRt dt + c̃ + µdt(ay + m̃ + s̃) + ”t dte≠fldt Vt+dt (a + ht s̃, m̃, 0)
⁄
Vt (a, m, s) = max +‰dt fl+⁄
f (m̃) + ⁄dtŴt (a, m̃, s̃)
c̃,m̃,s̃
+ [1 ≠ (µ + ”t + ‰ + ⁄) dt] e≠fldt Vt+dt (a, m̃, s̃)
s.t.c̃ + m̃ + s̃ Æ m + s
c̃, m̃, s̃ Ø 0
We focus on the parameter space where consumption allocation before agents exit the market
is always 0. Thus, c̃ = 0, and m + s = m̃ + s̃ = m0 , Since m0 is a constant,
ˆVt (a,m,s)
ˆs
= Rt dt,
Ut (a, m̃, m0 ≠ m̃) does not depend on m̃ given w, and m̃ is solved by the following problem:
max ”t dte≠fldt Vt+dt (a + ht (m0 ≠ m̃), m̃, 0) + ‰dt
0Æm̃Æm0
⁄
f (m̃) ≠ Rt dtm̃
fl+⁄
FOC
1
ˆ
”t ≠ht ˆa
+
ˆ
ˆm
2
⁄
Vt+dt (a + ht (m0 ≠ m̃), m̃, 0) + ‰ fl+⁄
f Õ (m̃) ≠ Rt + “m̃Ø0 ≠ “m̃Æw = 0
ˆ d
V
(a + ht (m0 ≠ m̃), m̃) =
ˆa t+dt
d
Ut (a, m, 0) =
da
=
ˆ
0
ˆ
0
Œ5
6
d
µy + ⁄ Ut+s (aÕ , mÕ , 0) e≠(µ+⁄+‰+fl)s ds
da
Tt (b) 5
µy + fi
6
µy
e≠(‰+fi+µ+fl)· d·
fl+‰+µ
Ë
È
µy
1 ≠ e≠(fl+fi+µ+‰)Tt (b)
fl+‰+µ
60
ˆ d
Vt+dt (a + ht (m0 ≠ m̃), m̃)
ˆa
ˆ Œ5
1
26
µy
≠(fl+fi+µ+‰)Tt+s (bt (a+(1+ht )(w≠m̃),m̃))
=
µy + ⁄
1≠e
e≠(µ+⁄+‰+fl)s ds
fl+‰+µ
0
ˆ d
Vt+dt (a + ht (m0 ≠ m̃), m̃)
ˆm
ˆ Œ5
6
ˆ
⁄
=
µ+⁄
Ut+s (a + ht (w ≠ m̃), m̃, 0) + ‰
f Õ (m̃) e≠(µ+⁄+‰+fl)s ds
ˆm
fl+⁄
0
ˆ
⁄
Õ
Œ
µ + ‰ fl+⁄ f (m̃)
+⁄
=
f Õ (m̃ + bt+s (a + ht (w ≠ m̃), m̃)) e≠(µ+⁄+‰+fl)s ds
µ+⁄+‰+fl
0
Haircut and indifference condition
Vtd (a, m)
Œ5
6
⁄
=
µ(ay + m) + ⁄Ut+s (a, m, 0) + ‰
f (m) e≠(µ+⁄+‰+fl)s ds
fl+⁄
0
⁄
µ(ay + m) + ‰ fl+⁄
f (m) ˆ Œ
=
+
⁄Ut+s (a, m, 0)e≠(µ+⁄+‰+fl)s ds
µ+⁄+‰+fl
0
ˆ
S
W ŝt+· Rt+· +
U
d (a
”t+· Vt+·
Vt (a, m̂t , ŝt ) =
ˆ
Œ
e≠
´·
”t+u du≠(µ+‰+⁄+fl)·
0
0
T
+ ht+· ŝt+· , m̂t+· ) + µ(ay + m0 ) X
⁄
+‰ fl+⁄
f (m̂t+· ) + ⁄Ut+· (a, m0 )
d·
V
Haircut must be such that, given the optimal choice (m̂t , ŝt ),
Vt (a, m̂t , ŝt ) = Vtd (a + ht ŝt , mt )
Initial borrowing of investors with LT projects
Ut (a, m, s) = maxa
0ÆbÆ h
t
Ó
Rt sdt + f (m + s + b)
1
2
1
2È
´·
´·
´ T (b) Ë
µay
+ 0t
‰ · 0 + µ ay ≠ be 0 Rt+s ds + fi fl+µ+‰
≠ be 0 Rt+s ds e≠(‰+µ+fi+fl)· d·
´·
where Tt (b) = inf · : be
ing problem:
0
Rt+s ds
Ô
ht+· = a . The maximization problem is equivalent to the follow-
61
max
a
0ÆbÆ 1+h
t
´ T (b)
≠(‰+µ+fi+fl)· d·
f (m + s + b) + 0 t fl+µ+‰+fi
fl+µ+‰ µaye
´
´ T (b)
·
≠b 0 t (µ + fi) e 0 Rt+s ds e≠(‰+µ+fi+fl)· d·
´·
Rt+s ds
´·
Rt+s ds
A sufficient condition for equation be
´·
e
0
Rt+s ds
h· is monotonically increasing. e
Market-clearing condition
0
0
=
Ë
a
ht+·
to have at most one solution is that
È
Rt+· ht+· + ḣt+· > 0.
ˆTt (b)
ht+·
È
=≠ Ë
ˆb
b Rt+· ht+· + ḣt+·
The density of deactivated lenders with portfolio (a + s· (1 + h· ), m· , 0) is ”· n0· e≠(µ+⁄)(t≠· ) . The
measure of active lenders at t is denoted n0t .
Demandt =
ˆ
0
Œˆ Œ
0
⁄”t≠· ≠s n1t≠· ≠s e≠(µ+⁄)s bt≠· ≠s (a + st≠· ≠s ht≠· ≠s , mt≠· ≠s , 0) e(R≠fi≠µ)·
I {· Æ Tt≠· (bt≠· ≠s (a + st≠· ≠s ht≠· ≠s , mt≠· ≠s , 0))} dsd·
ˆ Œ
+
⁄n1t≠· bt≠· (a, m0 , 0) e(R≠fi≠µ)· I {· Æ Tt≠· (bt≠· (a, m0 , 0))} d·
0
Supplyt = n1t ŝt
Default rate
Updating ”t depends on the measure of demand from borrowers and the flow of demand from
defaulting borrowers.
”t =
Demand of defaulting borrowert
Demandt
The distribution of the lender’s portfolio, F (a, m, s), a function of timing of counterparty default.
´·
´Œ
n1t = 0 ÷e≠(µ+⁄)· ≠ 0 ”t≠s ds d· . The measure of agents in the default cohort · (· < t) at moment
t is ”· n1· e≠(µ+⁄)(t≠· ) . And agents in the default cohort · have portfolio, (a + s· (1 + h· ), m· , 0).
62
Numerical algorithm to compute the transition path
Suppose that the shock to expectation arrives at t = 0.
1. Guess a sequence of interest rate, haircut and default rate, {Rt , ht , ”t } ’tØ0 .
2. Given the sequence, {Rt , ht , ”t } ’tØ0 , solve for the policy functions and value functions of
agents on the transition path.
3. Given the h0 , solve for the mass of initial default. And then, given the policy function of
agents, solve the distribution of agents along the transition path.
4. Given the distributions of agents and the policy functions, update the default rate of borrowers.
5. Given the distributions of agents and the policy functions, solve for the net demand of repo
borrowing. Update the interest rate according to the net demand of repo borrowing.
6. Given the value functions of agents, update haircut.
7. Go back to step 2 with the updated sequence of interest rate, haircut and default rate, until
convergence.
63
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